Physics-Guided Deep Network for Milling Dynamics Prediction

Kunpeng Zhu; Jun Li

doi:10.1016/j.eng.2024.09.027

Engineering ›› 2025, Vol. 55 ›› Issue (12) :71 -85. DOI: 10.1016/j.eng.2024.09.027

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Physics-Guided Deep Network for Milling Dynamics Prediction

Kunpeng Zhu ^a^,^b^,^*
, Jun Li ^a^,^b

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Abstract

Milling force is key to the understanding of cutting mechanism and the control of machining process. Traditional milling force models have limited prediction accuracy due to their simplified conditions and incomplete knowledge contained for model construction. On the other hand, due to the lack of guidance from physics, the data-driven models lack interpretability, making them challenging to generalize to practical applications. To meet these difficulties, a deep network model guided by milling dynamics is proposed in this study to predict the instantaneous milling force and spindle vibration under varying cutting conditions. The model uses a milling dynamics model to generate data sets to pre-train the deep network and then integrates the experimental data for fine-tuning to improve the model’s generalization and accuracy. Additionally, the vibration equation is incorporated into the loss function as the physical constraint, enhancing the model’s interpretability. A milling experiment is conducted to validate the effectiveness of the proposed model, and the results indicate that the physics incorporated could improve the network learning capability and interpretability. The predicted results are in good agreement with the measured values, with an average error as low as 2.6705%. The prediction accuracy is increased by 24.4367% compared to the pure data-driven model.

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Keywords

Milling force / Dynamics / Physics-guided network / Prediction

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Kunpeng Zhu, Jun Li. Physics-Guided Deep Network for Milling Dynamics Prediction. Engineering, 2025, 55(12): 71-85 DOI:10.1016/j.eng.2024.09.027

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1. Introduction

Milling has been one of the most applied machining processes for its excellent capability to create efficiently precision complex parts with wide range of materials. Milling force is an important factor that influences the milling process [1]. By predicting the milling force, the process parameters can be timely adjusted, tool wear and vibration may be suppressed, and the production may be carried out in reasonable milling conditions, which is of great significance. Accurate modeling helps to conduct parameter optimization and stability analysis more effectively [2]. The current milling force prediction models include physical, data-driven, and hybrid models.

The commonly used physical models include empirical formula based on experiments [3], [4], analytical method based on mechanics [5], [6], and mechanical model based on cutting load theory [7]. The mechanical model is more widely applied than the two others due to its capability to model instantaneous force, yet its real-time performance still needs improvement. Thus some researchers have developed models to reconstruct milling force based on the physical conversion relationship between force [8], [9], vibration [10], [11] and their fusion [12], [13]. Unfortunately, the generalization is still limited under unknown working conditions for these physical models. Altintas et al. [14] pointed that although the understanding of machining dynamics has increased significantly during the last six decades, the accuracy of dynamics predictions still suffers from measurement uncertainties, nonlinearities in the machine structure and process, and time-varying dynamics of machine tools and parts.

Data-driven models are developed for machining process modeling [15], which can directly extract features and patterns from the data without the need for complex mathematical modeling and understanding of process dynamics. They can learn the complex structural dynamics through machine learning methods [16] and address the highly uncertain issues in machining [17]. Some researchers have established mappings from current to force [18], [19], vibration to force [20], and milling condition to force [21], [22], eliminating the need for physical modeling and frequency response function testing. Unlike physical models, data-driven models usually contain more factors that affect milling force through the mapping relationship. However, it’s difficult to apply them in practical tasks as the poor interpretability arises from the lack of clear mechanisms [23]. Moreover, the lack of effective training samples typically leads to limited generalization.

To address the limitations of physical and data-driven models, hybrid models have received increasing attention. Navarro-Devia et al. [24] pointed out that the hybrid models combining Artificial Intelligence (AI) algorithms with physics may lead to promising solutions for milling dynamics problems. Zhong et al. [25] highlighted that traditional machine learning may be interjected with human domain expertise to help improve human-machine interactions and communications. It has been proved experimentally that the integrated approach fused theoretical analysis with big data can significantly improve the control accuracy of the machine tool system [26]. Attia et al. [16] remarked that hybrid models of physical and data-driven models provide further flexibility to handle modeling problems in machining fields. There are three key approaches to hybrid modeling.

(1) Embed prior knowledge through a specially designed data-driven model structure. Xu et al. [27] designed the network output as a vector sum of the cutter elements' forces, obtaining approximate forces that closely match the measured values. Agarwal et al. [28] concatenated the mechanical model after the multilayer perceptron for force estimation, avoiding the coefficient determination. Xie et al. [29] replaced complex computations with convolutional layers on integrated geometric machining information images and produced high-precision milling force prediction. Zhu et al. [30] proposed a Fourier filtering demodulation for adaptive component detection, improving bearing fault classification accuracy. Li and Zhang [31] stacked recurrent neural network (RNN) unit layers to iteratively compute the residuals of the linearized state-space equation, enhancing the robustness of wind turbine fatigue damage evaluation. Hanachi et al. [32] used regularized particle filtering to weigh the tool wear prediction results of the empirical model and neural network, significantly calibrating the physical model errors. These models effectively improve the prediction performance, yet difficult to extend to more complex systems due to the specifically designed structures.

(2) Guide the learning of data-driven models with physics information. Rahimi et al. [33] employed self-excited vibration theory to guide the AlexNet model in distinguishing between forced vibrations and chatter at transient states, leading to only 1.1% error detection rate. Corson et al. [34] applied three physical models to establish the initial beliefs before testing to determine the probability of milling stability, significantly reducing the number of tests required to identify the optimal stable parameters. Xiong et al. [35] set the monotonicity of flow feature in the equipment degradation process as a physical constraint to control the generator to produce synthetic degradation trajectories of turbofan engines, enhancing the physics plausibility of the data. Lu et al. [36] used the variation of bearing feature indicators as dynamic weights to construct a loss function and improve the potential consistency of remaining lifespan estimation. They enhance the physical consistency of the results while limited by the relatively simplified knowledge.

(3) Augment data for data-driven models with physics information. Vaishnav et al. [37] trained the network with mechanical model samples, reducing the experimental needs. Finkeldey et al. [38] fused online signal features with tool-workpiece engagement information provided by geometric physics-based simulation, improving force prediction accuracy. Wang et al. [39] performed force transfer learning using simulated and experimental data as the source and target domains, demonstrating significant advantages over regular networks. Liu and Altintas [40] narrowed the gap between simulation and real dynamics by progressively learning finite element simulated and real data using a progressive network to estimate machine position-related dynamics. Ha and Fink [41] recently leveraged domain knowledge to generate faulty samples from healthy target data, achieving the cross-domain diagnosis of gearbox faults under sample scarcity. Tai and Altintas [42] trained the gated recurrent unit network by mathematically simulated vibration spectrums under various bearing faults, achieving a high-precision spindle fault diagnosis. Alberts et al. [43] claimed that simulated vibration data serves as a powerful tool to overcome the constraints of small-scale real-world data sets.

Table 1 summarizes the characteristics of the current approaches. Although the hybrid models integrate the strengths of physical and data-driven methods, they still face challenges regarding limited generalization and poor interpretability without universal and deep mechanisms. Their reliability and adaptability in practical applications may be compromised due to the neglect of explicit physics laws in the machining process [44]. Rezaei et al. [45] pointed out that there is a pressing need for an approach that can accurately predict milling dynamics using limited experimental data. According to Gao et al. [46], integrating physical and data-driven approaches can lead to model interpretability that better aligns with the requirements of the modern manufacturing industry. With this idea, a physics-guided data-driven model is proposed in this study. Its characteristic lies in the utilization of a physical loss function that incorporates dynamics knowledge, and the joint training with pre-training samples and experimental data. This method aims at the inherent limitations of low generalization and lack of interpretability in data-driven models due to the absence of external guidance. The key contributions of this study include:

(1) Developing a novel deep learning-based method for milling dynamics modeling, which can predict the milling force and cutter-spindle system vibration simultaneously under different milling conditions.

(2) Proposing a pre-training strategy with samples generated by the milling dynamics model, which augments the training samples and improves the model performance.

(3) Proposing a physically meaningful loss function as the constraint, enhances the interpretability and the physical consistency of the model.

This paper is organized as follows. Section 2 describes the milling dynamics process and proposes the physics-guided data-driven model. The proposed model is experimentally verified and analyzed for its validity in Section 3. Section 4 summarizes the conclusions.

2. Physics-guided data-driven model for milling dynamics prediction

2.1. The milling dynamics

Milling dynamics is the process of interaction between the milling force and relative displacement of cutter and workpiece. It is complex and has nonlinear characteristics due to the intermittent cutting nature. In the two-dimensional (2D) planar milling system (comprising the normal direction X and feed direction Y) illustrated in Fig. 1(a), the relative displacement caused by vibration (hereafter referred to as displacement) can be decomposed into instantaneous uncut chip thickness through spatial coordinate transformation, which affects the milling force. In the milling system, the force acts on the elastic damping system as an excitation, causing vibration and changing the relative displacement [47]. The process dynamics can be characterized by differential Eq. (1)

(1)

M \overset{. .}{q} (t) + C \dot{q} (t) + Kq (t) = F_{s} (t) + F_{d} (t)

where

M = [\begin{matrix} m_{x} \\ m_{y} \end{matrix}]

C = [\begin{matrix} c_{x} \\ c_{y} \end{matrix}]

, and

K = [\begin{matrix} k_{x} \\ k_{y} \end{matrix}]

are the mass, damping, and stiffness matrices, respectively.

q (t) = {[q_{x} (t), q_{y} (t)]}^{T}

is the displacement vector in X and Y directions, T is the transpose operator.

F_{s} (t) = {[F_{x s} (t), F_{y s} (t)]}^{T}

is the static milling force vector based on rigid-body kinematics, computable via the dual-mechanism mechanical model [6].

F_{d} (t) = k_{c} [q (t) - q (t - T)] {[\sin ϕ, \cos ϕ]}^{T}

is the dynamic milling force vector based on the regeneration effect. T is the delay period standing for the time passed after the last edge cut. ϕ is the cutter rotation angle.

k_{c} = {[k_{tc}, k_{rc}]}^{T}

is the milling force coefficient standing for the shearing mechanism.

For the large milling tool as used in most milling systems, the vibration is significant in the normal direction X and feed direction Y while small and negligible in the axial direction. Therefore, only the 2D vibration is analyzed in this study.

The Eq. (1) describes the coupling effect between the milling force and the vibration with a certain degree of accuracy. Generally, the numerical integration method can be used to solve the equation, enabling the simulation of milling dynamics in the time domain, as depicted in Fig. 1(b). However, the physical model simplifies the dynamics process. In contrast, data-driven models can learn knowledge from measured signals and achieve more precise modeling for the process. However, their black-box nature leads to poor interpretability compared to physical models. Furthermore, data-driven models typically rely heavily on the abundance and diversity of actually collected signals. To address the issues of poor interpretability and insufficient samples, a physics-guided data-driven model is developed in this study.

2.2. The physics-guided deep neural network (DNN)

DNN is used as the base data-driven learner in this study, owing to its strong nonlinear mapping ability and flexible structure. The proposed model is shown in Fig. 2, which consists of two modules: a DNN model and a milling dynamics model. The DNN predicts the milling force and displacement according to inputted milling parameters and time. The dynamics model generates samples similar to actual signals for network pre-training and supplies the delay differential equation as a strict constraint to compel predictions to adhere to physical consistency.

Pure data-driven DNN exhibits two limitations. The first is that insufficient training samples may lead to low generalization. Therefore, simulation is utilized for expanding the samples, which generates samples comprising milling force and corresponding spindle displacement within similar conditions by the dynamics model identified through the experimental data. These samples can pre-train the network and enhance its capability to learn general knowledge. Then experimental data are used to fine-tune the network parameters to reduce the learning burden on specific tasks once pre-training is completed.

The second is that the typically adopted mean square error (MSE) loss function in pure data-driven DNN may lead to poor interpretability and physical inconsistency [48]. Minimizing the MSE amounts to label fitting, which lacks interpretability and may lead to results inconsistent with the objective physical rules under noise disturbance. Therefore, the relationship between force and vibration signals is used as a physical constraint during the training process.

2.2.1. The modified DNN model for milling force modeling

The DNN structure is shown in Fig. 3, which contains 1 input layer, 8 hidden layers, and 1 output layer. The input consists of the milling parameters n, f_z, a_p, a_e, and time t, which determines the milling force at a specific instant.

The milling force can be decomposed into static force F_s (independent of vibration) and dynamic force F_d (related to vibration), and the ultimate predicted force value consists of the sum of these two parts:

(2)

F = F_{s} + F_{d}

The inputs are simplified by structuring the same set of milling parameters and time sequence as a matrix:

(3)

I (k) = [n (k), f_{z} (k), a_{p} (k), a_{e} (k), t (i)], i = 1, 2, . . ., L

where k represents the index of different milling conditions, and L is the time length.

The milling force is decomposed into F_s and F_d in the last hidden layer of the network, which and the dynamic displacement serve as the output of the hidden layer, denoted as latent variables, and expressed as follows:

(4)

L (k) = [F_{x s} (i), F_{y s} (i), F_{x d} (i), F_{y d} (i), q_{x} (i), q_{y} (i)], i = 1, 2 . . ., L

The output is also a matrix:

(5)

O (k) = [F_{x} (i), F_{y} (i), q_{x} (i), q_{y} (i)], i = 1, 2, . . ., L

Both batch inputs and batch outputs are in the form of 3rd-order tensors, as shown in Fig. 3, where each layer corresponds to a set of milling conditions. In which, samples at arbitrary delay time T can be indexed through interpolation. The input data are mapped into the dynamic force and static force as latent values through the autoencoder structure embodying the regenerative effect, then summed to the overall force.

2.2.2. The physics guidance for the DNN

The guidance of physics knowledge is achieved in two ways in the model. First, the simulated data provides general information for initial DNN model learning. Second, the explicit physical equation provides a strict constraint to DNN.

The MSE loss function is commonly applied to evaluate the degree of inconsistency between the DNN output values and the actual values, formulated as:

(6)

{Loss}_{mse} = \frac{1}{4 L} \cdot \sum_{i = 1}^{L} {[\hat{O} (i) - O (i)]}^{2}

where

\hat{O} (i)

and

O (i)

are the predicted and actual observed values at time i, respectively.

However, it notably lacks underlying mechanisms of the real process. To address this limitation, a physics-guided loss function is proposed as the regularization constraint for DNN to ensure the consistency between the prediction results and physics knowledge:

(7)

{Loss}_{phy} = {Loss}_{p x} + {Loss}_{p y}

where

{Loss}_{p x}

and

{Loss}_{p y}

represent the loss arising from physical inconsistency between the predicted results in two directions and the vibration differential equation, as follows:

(8)

{Loss}_{phy} = MSE (M \ddot{q} + C \dot{q} + Kq - F_{s} - F_{d})

The right part of Eq. (7) expresses the residual force. Considering that different loss terms must be dimensionally consistent, the following transformation is used to unify the dimensions of physical losses:

(9)

{Loss}_{phy} = \frac{{Loss}_{p x}}{S_{F x}} + \frac{{Loss}_{p y}}{S_{F y}}

Among them,

S_{Fx}

and

S_{Fy}

are the scales on the F_x and F_y dimensions when normalizing samples, respectively.

Finally, the loss function of the DNN comprises the MSE component and the physical component:

(10)

Loss = {Loss}_{mse} + λ_{phy} {Loss}_{phy}

where

λ_{phy}

is the hyperparameter representing the proportion of physical loss in the overall loss:

(11)

λ_{phy} = 1 - \exp (- \frac{{Loss}_{mse}}{{Loss}_{phy}})

The hyperparameter

λ_{phy}

reflects that DNN focuses on the fitting of labels at the initial stage of training, with a progressive inclination toward the constraints imposed by the physical loss function as the number of training epochs increases.

Considering that discrete data cannot be subjected to continuous differentiation methods, the two-point difference approximation derivative method is employed to compute the physical loss in DNN:

(12)

\dot{q} = q^{T} D f_{s}

(13)

q^{T} = f_{s} D^{2} q^{T}

where

f_{s}

is the sampling frequency, and

D

is the tensor differentiator defined as follows:

(14)

D = {[\begin{matrix} - \frac{3}{2} \\ 2 & - \frac{1}{2} \\ - \frac{1}{2} & 0 \\ \frac{1}{2} & ⋱ & - \frac{1}{2} \\ 0 & \frac{1}{2} \\ \frac{1}{2} & - 2 \\ \frac{3}{2} \end{matrix}]}_{(L \times L)}

To maintain the physical significance of loss terms, it is necessary to reverse-normalize the normalized data to the original scale when computing the physical loss.

2.2.3. The physics-guided DNN training

The training procedure consists of three stages: data preparation, pre-training, and fine-tuning.

In the data preparation stage, the milling dynamics model is identified based on experimental data. Then multiple combinations of milling parameters within a similar range to the experimental condition are substituted to the known dynamics model to generate simulated data. The overall data set consists of simulated data and experimental data.

DNN is initially pre-trained using simulated data, with the first several layers' parameters frozen to preserve general mapping relationships. Fine-tuning is then applied to the pre-trained model for further optimization with experimental data. Note that the model hyperparameters need to be determined before training begins.

There are six hyperparameters in this model: the number of network layers, the number of nodes in each layer, activation function, maximum training epochs, frozen layers, and learning rate. While there are no specific rules for hyperparameters setting, the optimal configuration can be chosen based on the convergence speed of the loss and the test accuracy. The determination of activation function and learning rate can be guided by some prior knowledge.

The leaky rectified linear unit function is used as the activation function in this study to prevent the gradient from disappearing while ensuring neuron weights updates. As illustrated in Fig. 4, it achieves a nonlinear mapping relationship through a linear function with slope

β

when the weighted and biased input is negative.

Moreover, the weights and bias of the network are randomly initialized with different seeds to achieve better performance. During model training, the adaptive learning rate technique is employed to dynamically adjust the learning rate according to gradient information, ensuring optimal parameter updates.

The model training procedure is summarized in Algorithm 1.

Algorithm 1 The training procedure of physics-guided DNN.

Stage 1: data preparation

1: Identify the system parameters by the milling dynamics model

2: Input milling parameters similar to the processing conditions

3: Generate training samples using the known dynamics model

Stage 2: pre-training

4: Initialize network parameters with random seeds

5: Train the initial DNN model with simulated data

6: Freeze the weights and bias of the first several layers

Stage 3: fine-tune

7: Call the pre-trained model in Stage 2

8: Re-training the pre-trained model with experimental data

Parameters updates process

9: for

t = 0

: max train epochs do

10: Compute

Loss

11: Compute gradient

δ

12: Update

W \to W_{n}

with

δ

13: Update

B \to B_{n}

with

δ

3. Experimental verification and result discussions

3.1. Experimental setup

To verify the proposed method, experiments are carried out on a three-axis high-speed vertical machining center DX-650 produced by the Chinese Liangjiang Company (China). The experimental setup is shown in Fig. 5 and corresponding equipment details are shown in Table 2. The whole set of measurement equipment mainly includes a force measuring table, a vibration collecting device, and a charge amplifier. The Swiss Kistler9129A dynamometer (Kistler) is used to acquire milling forces in the X and Y directions in real-time during the milling process. The measuring range of the dynamometer is ±10 kN and the accuracy is 0.001 N. The Swiss Kistler8763B vibration sensor (Kistler) is used to acquire the vibration signals in the milling process of the spindle. The experimental data acquisition method is multi-channel timing acquisition, and the sampling frequency is set to 50 kHz. The data acquisition card uses the Slovenian Dewesoft SIRIUsi-HS 8-channel module (Dewesoft, Slovenia) to acquire and store signals. The tool used is the 4-flute carbide flat-end milling cutter with a diameter of 6 mm and a helix angle of 48°. The material of the workpiece is 42CrMo, and the size is 100 mm × 90 mm × 30 mm. Additionally, the tool type and workpiece material are fixed to conduct research within a specific scope.

The process parameters with spindle speed, feed rate, milling depth, and milling width are selected and combined according to the milling process parameter manual under a recommended range. Since high material removal rate may lead to machining chatter and tool damage, while a low value may result in a lower signal-to-noise ratio. The values of four parameters are set as (5 000, 7 500, 10 000), (800, 1200, 1600), (1.2, 1.4, 1.6), and (1.2, 1.4, 1.6). Each parameter is combined separately and simplified to 9 groups using the Taguchi method. As shown in Table 3, a set of orthogonal experiments with four factors and three levels of milling parameters are designed to cover a relatively complete working condition. The milling force and cutter-spindle vibration signals are collected synchronously. However, data acquisition in actual processing tasks is often limited to part of working conditions, with others unknown. The data under three groups of working conditions are set as known to align the actual situation.

3.2. Identification of milling system dynamics

First, it’s necessary to preprocess the raw data collected from sensors. The raw signals of cutter-spindle vibration acceleration are processed using the five-point cubic smoothing filtering method to eliminate high-frequency random noise. Then, the time-domain signal is transformed into a frequency-domain signal through the discrete Fourier transform, and the acceleration signal is double integrated in the frequency-domain to obtain the cutter-spindle displacement. The denoised signals are resampled at intervals of 1° of tool rotation and segmented into multiple sequences of 0.01 s each, which are used for learning.

The data used for pre-training is generated using dynamic time-domain simulation. To effectively modify the DNN parameters through pre-training, it’s necessary to minimize the differences between simulated data and experimental data as much as possible. In this study, the parameters of the experimental data, including milling force coefficients, radial runout length and angle, initial immersion angle and vibration parameters, are identified first, and then simulated data are generated using these parameters.

3.2.1. Identification of milling force coefficients

According to the dual-mechanism mechanical model shown in Fig. 1(a), the milling force at time i is computed below [7]:

(15)

F_{s} (i) = A (i) k_{m}

Given the initial immerging angle of the force signal, radial runout angle, and runout length, the error between the modeled and actual force is as follows:

(16)

e (ϕ_{0}, r, γ) = ∥A (ϕ_{0}, r, γ) {[A (ϕ_{0}, r, γ)]}^{†} F_{m} - F_{m}∥

where

†

represents the pseudo-inverse operation.

Any sequence of milling force can be considered as the observed value with its milling force coefficients serving as the latent variable. Therefore, the milling force is modeled as the state-space model as follows and its milling force coefficients can be identified through the Kalman Filter method:

(17)

\{\begin{matrix} k_{m} (i + 1) = ψ (i) k_{m} (i) + ω (i) \\ F (i) = H (i) + v (i) \end{matrix}

where

k_{m} = {[k_{tc}, k_{rc}, k_{te}, k_{re}]}^{T}

is the milling force coefficient vector consists of shear force coefficients and edge force coefficients in tangential and radial directions.

ψ

and

H

are transition matrix and observation matrix, respectively.

ω

and

v

are process noise and measurement noise, respectively.

3.2.2. Identification of vibration parameters

The single-degree-of-freedom vibration system can be modeled as an autoregressive model [49] with 2 autoregressive terms (AR-2).

(18)

q (i) - \sum_{l = 1}^{2} a_{l} q (i - l) = b_{0} F (i)

The AR-2 model is fitted to measured data using the Levenberg–Marquardt method, with physical parameters derived from its relationship with model coefficients as follows:

(19)

\{\begin{matrix} a_{1} = Rev [(M + Δ t C + {(Δ t)}^{2} K] \cdot (2 M + Δ t C) \\ a_{2} = Rev [(M + Δ t C + {(Δ t)}^{2} K] \cdot M \\ b_{0} = Rev [(M + Δ t C + {(Δ t)}^{2} K] \cdot {(Δ t)}^{2} \end{matrix}

where

Rev

[⋅] represents the operator for computing each element reciprocal in a matrix,

Δ t

is the time step.

The error in fitting the AR-2 model is computed below:

(20)

ε (a_{1}, a_{2}, b_{0}) = \sum_{i = 1}^{L} q (i) - a_{1} q (i - 1) - a_{2} q (i - 2) - b_{0} F_{m} (i)

The algorithm of the identification program is shown in Algorithm 2.

Algorithm 2 Identification of parameters.

Input:

F_{m}

q_{m}

Output:

Φ_{0}

r

γ

k_{m}

M

C

K

1: Initialize

Φ_{0} = 0

r = 0

γ = 0

k_{m} = {[0 0 0 0]}^{T}

2: for

Φ_{0} = 0 : 2 π

3: for

r = 0 : \frac{R}{2}

4: for

γ = 0 : π

5: Compute

e (Φ_{0}, r, γ)

as Eq. (16)

Φ_{0}, r, γ = \underset{Φ_{0}, r, γ}{\arg \min [e (Φ_{0}, r, γ)]}

7: Compute the coefficients

k_{m}

by Kalman Filter method

8: Initialize

a_{l} = {[\begin{matrix} 0 0 \\ 0 0 \end{matrix}]}^{T}

b_{0} = {[\begin{matrix} 0 0 \\ 0 0 \end{matrix}]}^{T}

9: for

i = 3 : L

10: Compute

ε (a_{l}, b_{0})

as Eq. (20)

11: Search

a_{l}

b_{0}

by Levenberg–Marquardt algorithm

12: Compute

M

C

K

according to Eq. (19)

The dynamics model parameters are shown in Table 4, which is used to generate simulated samples for pre-training.

3.3. Model specifications and training

The milling conditions for generating simulated data are artificially selected and combined, following two criteria:

(1) On the premise that the selected milling condition is similar to the actual working condition, the parameters combination shall be diverse.

(2) Recommended values are selected according to the milling process parameter manual to avoid damage to the milling tool and machine tool.

It is worth noting that the different time lengths of the training samples can influence the time required for training error convergence. Considering the periodicity of force, input t is only taken one-period length of tool rotation for each parameter combination to conserve computational resources. By substituting them into the dynamics model with the identified parameters, simulations can be conducted in the time domain based on Eq. (1). Specifically, at each time step, the cutting load for each element of the cutter (product of the chip thickness and force coefficient) is computed, and then weighted accordingly. Subsequently, the forces and vibrations are updated according to Fig. 1.

Six sets of corresponding forces and displacements are generated, each of which includes forces and displacements in the X and Y directions within a period. Fig. 6 exhibits a favorable agreement between the estimated values and the concurrently collected actual signals under different cutting parameters, both in terms of amplitude and trend, which indicates that the identified dynamic parameters are within a reasonable range.

Due to the significant differences in data scale across milling parameters and time, the pre-training data are normalized and standardized before training.

Through extensive comparative experiments, the hyperparameters are adjusted and ultimately determined, as detailed in Table 5.

3.4. Results and discussion

This section demonstrates the predictive performance of the proposed model, explores the interpretability of the DNN model, and compares the performance of different models. The following cross-validation experiments are conducted to validate the model. Simulation data from six conditions are used for pre-training, two sets of experimental data are used for fine-tuning, and one set is used for validation.

3.4.1. Predicted results of the proposed model

To evaluate the accuracy of the proposed model, the predicted milling forces and displacements, as well as their measured values, are compared under the three test groups of experimental milling parameters. Fig. 7 demonstrates the comparative assessment of milling forces and displacements predicted using the proposed model with experimentally measured signals. It can be seen that the red and blue lines match well, indicating that the predicted results of the model are in good agreement with the measured values, and the model can accurately predict the instantaneous values from the dynamic changes in milling time.

It suggests that the model has successfully learned the underlying dynamics of the milling system, enabling it to predict the change of dynamic performance of the system under unknown working conditions according to known working conditions. In contrast to milling forces, the proposed model demonstrates superior accuracy in predicting displacements. This discrepancy arises from that the variation in the displacement signal exhibits a relatively smooth and gradual trend while the force signals exhibit more intricate periodic patterns.

It can be observed that the model performs remarkably well for No.1 and No.3, while slightly lower for No.7. The maximum error occurs in the forces of No.7 test data. It can be attributed to the substantial increase in rotational speed under that condition, which changes the dynamics characteristic and leads to an increase in the frequency and complexity of the signals. This results in multiple peaks at certain crest positions, complicating the fitting process. While the signals under No.1 are more stable, followed by condition No.3, which exhibits better results.

The accuracy of the predicted results is quantitatively evaluated via three criteria: mean absolute error (MAE), mean absolute percentage error (MAPE), and root mean squared error (RMSE). Smaller values for each criterion correspond to better model performance.

(21)

MAE = \frac{1}{L} \sum_{i = 1}^{L} | o (i) - \hat{o} (i) |

(22)

MAPE = \frac{100 %}{L} \cdot \sum_{i = 1}^{L} |\frac{\hat{o} (i) - o (i)}{o (i)}|

(23)

RMSE = \sqrt{MSE}

All the errors are listed in Table 5. The MAPE of the proposed model is 1.6585%, 2.6736%, 1.5506%, and 1.3392% for the four predictive variables, indicating that the proposed model can make relatively accurate predictions of milling force and displacement.

To evaluate the effect of the physical loss function constraint, the following comparisons are made: the predicted displacements are substituted into the vibration differential equation to derive the corresponding forces, and compared with the predicted values of the proposed model, as shown in Fig. 8.

It is evident that the derived forces and predicted values exhibit a consistent trend, revealing that the predicted forces conform to the principle of the vibration differential equation. There are localized values of misalignment, which can be attributed to computational biases inherent in the model. However, the errors are negligible compared to the actual values and remain within an acceptable margin. It indicates that the results exhibit a high level of physical consistency.

To gain a better understanding of the effects generated by the physics mechanisms including pre-training and physical loss function, an ablative study was conducted where different components of the model were removed. Four comparative cases achieve it: pure DNN, only employs pre-train, only employs physical constraint, and employs both pre-train and the physical constraint (i.e., the proposed model). Fig. 9 illustrates this result.

The comparison reveals that the error under physical constraint (10.5613%) slightly exceeds the error under pre-training (9.4520%), indicating pre-training surpasses physical loss constraint in boosting network performance. This aligns with the nature that the amount of data has a more significant impact on learning efficiency than other factors. Integrating the physical constraint with the pre-training strategy yields remarkable enhancements in prediction accuracy.

3.4.2. The interpretability of the proposed model

A common method to interpret model learning is visualization of model parameters [51]. The basic structures of the DNN model are neurons (as shown in Fig. 10), where the parameters are the weights and bias of each neuron. The network training process involves iterative updating of these parameters, obeying the following rules:

(24)

\{\begin{matrix} W_{a, b}^{j + 1} = W_{a, b}^{j} + α \cdot \frac{\partial ∂Loss}{\partial {∂W}_{a, b}} \\ B_{a, b}^{j + 1} = {∂B}_{a, b}^{j} + α \cdot \frac{\partial ∂Loss}{\partial B_{a, b}} \end{matrix}

where α represents learning rate.

The DNN achieves the goal of minimizing loss by searching for the optimal parameters in training. Therefore, loss plays a guiding role in updating model parameters in two ways in this study: the guidance of simulated data as labels and the guidance of physical loss function. To visually elucidate the interpretability of the proposed model, the evolution of loss, neuron weights, and biases of the proposed model are compared with those of the traditional DNN throughout the training process.

The evolutions in loss values during iterative updates through Eq. (10) are depicted in Fig. 11. The loss converges faster than traditional DNN to a sufficiently low level (< 0.04) in the initial 2000 iterations, owing to that part of the parameters have reached the optimal values after pre-trained. In subsequent stages, the average level of loss in the proposed model is lower than that of the traditional DNN. During this stage, there is an oscillation and convergence of the physical loss, which indicates that the model is overcoming the inconsistency that violates the physical constraints repeatedly.

The proposed model exhibits a lower loss level compared to the traditional DNN. However, certain signals are notably influenced by noise values in specific samples. Particularly at higher rotational speeds, the noise patterns closely resemble the fluctuations in peak and valley positions within the signals. Both the traditional and proposed models produce higher loss levels on these samples.

Furthermore, the parameter variations of the third layer (with 4800 weights and 60 biases) and the fourth layer (with 2400 weights and 40 biases) are depicted in Fig. 12 for different training stages. The weights and biases exhibit characteristics similar to random noise with zero mean values in the initial model, and they change significantly after pre-training. However, parameters may reach the optimal values after being fine-tuned by the experimental data and don't diverge significantly from the pre-trained values. This reaffirms the effective guidance provided by simulated data.

The difference between the physics mechanisms used for generating training data and the physical constraint lies in their effect stages and objectives. The use of simulated data aims to enable the network to capture the underlying patterns in limited data and rapidly reach a proper initialization. Embedding a constraint that deviates from the dynamics mechanism when learning real data, ensures that the network learns to generalize well and is not simply memorizing the training data. The Eq. (25) further analyses the difference.

(25)

Loss = \{\begin{matrix} F_{net} (I) - f_{phy} (I), Pre-train \\ F_{net} (I) - O + λ_{phy} (F_{net} - f_{phy}), Fine-tuning \end{matrix}

in which,

F_{net}

represents the nonlinear relationship of the network, and

f_{phy}

represents the relationship of the dynamic equations. It is worth noting that during the pre-training stage, the

F_{net}

outputting results that match the label values may not be equal to

f_{phy}

. However, after incorporating the physical constraints, it is guided to progressively converge toward

f_{phy}

3.4.3. Comparison with different models

The mechanical model and traditional DNN model are two representative models widely used in milling force modeling, which are compared with the proposed model in this section. In addition, the proposed model is compared with two data-driven methods commonly used in nonlinear modeling, namely support vector regression (SVR) and Gaussian process regression (GPR). Furthermore, two representative advanced methods have been chosen for comparison with the proposed model. The first is physics-residual DNN (PRDNN), which sequentially connects the physical model with DNN to output the predicted results for forces and displacements. The second is the physics-weighted DNN (PWDNN), which weights the predicted outputs from the physical model and DNN to obtain the final predicted values.

The structure and parameters of each model are adjusted to achieve optimal performance. The mechanical model is the pure physical model based on Eq. (1) without parameter settings. The DNN has 7 hidden layers with (100, 80, 40, 40, 30, 40, 50) nodes, and shares the same other settings from the values specified in Table 5. The grid search method is applied to tune the parameters of other several compared models. Both SVR and GPR utilize the widely used Gaussian kernel. The regularization parameter for SVR is set to 20, with a 0.4 kernel bandwidth and a 0.01 fitting tolerance. As for GPR, the mean is set to 0 while the kernel bandwidth is 0.4. The PRDNN has 6 hidden layers with (100, 80, 50, 30, 30, 20) nodes. The PWDNN includes a module that is consistent with the DNN, and 2 additional layers with (20, 20) nodes for weighting the outputs of the physical model and the DNN. The test set is the three data groups for validation in Table 6.

Fig. 13 provides the comparison in No.7 about milling forces and displacements predicted using different models and their measured values. It can be observed that the physics-guided DNN model predicts the contour and magnitude of milling forces and displacements accurately in comparison to the other models. The deviations mainly lie in the peak and valley positions which are difficult to learn. The physical constraints in the proposed model are derived from physics prior knowledge obtained from known data. However, in high-speed milling operations, the vibration displacement is frequently affected by transient disturbances occurring at specific moments, such as material inhomogeneities. These disturbances can induce deviations in the system parameters identified using physics knowledge, thereby introducing biases in the established constraints. Consequently, the model may encounter challenges in accurately predicting these pilots.

To quantitatively compare the performance of different models, the MAPE on the predicted results of five models in the test set are listed in Table 7. The overall error of the proposed model is 24.4367% higher than that of the traditional model. It can be known that the proposed model exhibits significantly better performance than the other four models. Compared to the mechanical model, the proposed model utilizes data-driven techniques to capture factors not considered in the physical model. Compared to the traditional DNN model, physics-guided learning solves the lack of samples and ensures physical consistency. The proposed model integrates the advantages of the traditional physical model and data-driven model and shows good performance.

Remarkably, the mechanical model exhibits significant deviations in displacement prediction, which is due to the dynamic characteristics of the system potentially changing with variations in rotational speed. The physical model relying on known parameters is unable to forecast such dynamic behavior. The lower prediction accuracy of the traditional DNN, SVR, and GPR models can be attributed to the poor approximation of relationship by curve fitting technique owing to the presence of outliers and noise in the experimental data. The predicted values of the DNN, SVR, and GPR models exhibit overall consistency with the experimental values in terms of trends. However, there are significant local differences. In cases where samples are insufficient, the performance of data-driven models without guidance from physical knowledge is severely limited. Due to the kernel trick utilized in SVR and GPR, the two models exhibit worse than DNN. In comparison, the parametric model is more advantageous in learning the underlying physics laws of dynamics systems [50]. Based on the outcomes, it can be concluded that the physics-guided model realizes the relationship between cutting conditions and dynamics better in comparison to the traditional model, thereby enhancing the prediction accuracy of the traditional DNN model.

There are similar prediction performances on the PRDNN and PWDNN. Despite they fused the results from the physical model and DNN to improve the pure data-driven model, the results at the peaks and valleys, as well as the local rapidly changed positions are not accurate enough. The two models are sensitive to the initial assumptions made about the dynamics prior. The discrepancies between the model assumptions and the testing conditions can lead to distorted inference results. When the physical modeling lacks precision, the ultimate outputs mainly rely on the data-driven module. However, they are difficult to learn the physical consistency owning that the dynamics knowledge is not embedded into the data-driven module.

Unlike previous advanced methods, the proposed model that incorporates the dynamics of force and vibration displacement shows improvements in learning performance and understanding of the underlying mechanisms.

4. Conclusions

A novel physics-guided deep network for milling dynamics prediction is developed to address the challenges of incomplete knowledge in the construction of physical models and lack of interpretability in data-driven models. The following conclusions are achieved based on the validation with milling experiments:

(1) Pre-training strategy based on physical simulation embeds the prior dynamics knowledge into the network, accelerating convergence and optimizing initial model parameters. This strategy enhances the network's ability to efficiently learn and adapt to real milling dynamics.

(2) Furthermore, the loss function based on the vibration differential equation not only improves the physical consistency of predicted results but also enhances the interpretability of the traditional data-driven model.

(3) The two physics-guided approaches have jointly improved the accuracy of network prediction. The total average error is as low as 2.6705%, and overall prediction accuracy has been improved by 24.4367% compared to the pure physical model.

The proposed model can achieve cutting force prediction and guide further parameter optimization. The structure and design concept of this model is easy to generalize to other fields of the cutting process including turning and drilling dynamics, which can improve the interpretability of data-driven models.

It is worth noting that there are still many uncertain factors such as variation of tool wear in actual milling processing, which may cause deviations in the prediction. The real-time monitoring factors are warranted to be incorporated into the model in future studies to explore the full potential of this approach in relevant research fields and practical applications.

CRediT authorship contribution statement

Kunpeng Zhu: Conceptualization, Formal analysis, Funding acquisition, Methodology, Supervision, Writing - original draft, Writing - review & editing. Jun Li: Investigation, Methodology, Software, Writing - original draft, Writing - review & editing.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

This work was supported in part by the National Natural Science Foundation of China (52175528), in part by the National Key Research and Development Program of China, the Chinese Ministry of Science and Technology (2018YFB1703200).

Nomenclature

OXY$\ \ \ \ \ \ \ \ $ 2 D coordinate system in milling

M, C, and K $\ \ \ \ \ \ \ \ $ Mass, damping, and stiffness matrices

$\boldsymbol{q}, \dot{\boldsymbol{q}}, \text { and } \ddot{\boldsymbol{q}}$ $\ \ \ \ \ \ \ \ $ Displacement, velocity, and acceleration caused by vibration

$m_x$ and $m_y$ $\ \ \ \ \ \ \ \ $ The mass in X and Y directions

$c_x$ and $c_y$ $\ \ \ \ \ \ \ \ $ The damping in X and Y directions

$k_x$ and $k_y$ $\ \ \ \ \ \ \ \ $ The stiffness in X and Y directions

F $\ \ \ \ \ \ \ \ $ Overall milling force

F_m and q_m $\ \ \ \ \ \ \ \ $ Measured value of milling force and displacement

k_m $\ \ \ \ \ \ \ \ $ Milling coefficients

k_tc and k_rc $\ \ \ \ \ \ \ \ $ The coefficients standing for shearing mechanism in tangential and radial directions

k_te and k_re $\ \ \ \ \ \ \ \ $

The coefficients standing for ploughing mechanism in tangential and radial directions

T $\ \ \ \ \ \ \ \ $ Delay period of tooth pass

T $\ \ \ \ \ \ \ \ $ Transpose operator

$\Phi$ and $\Phi_{0}$ $\ \ \ \ \ \ \ \ $ Radial immersion angle of cutter and its initial value

n $\ \ \ \ \ \ \ \ $ Spindle speed

f and f_z $\ \ \ \ \ \ \ \ $ Feed rate and feed per tooth

a_p $\ \ \ \ \ \ \ \ $ Axial milling depth

a_e $\ \ \ \ \ \ \ \ $ Radial milling width

t $\ \ \ \ \ \ \ \ $ Milling time

k $\ \ \ \ \ \ \ \ $ Index of different milling conditions

i $\ \ \ \ \ \ \ \ $ Index of discrete time instant

Loss $\ \ \ \ \ \ \ \ $ Overall loss when training

L $\ \ \ \ \ \ \ \ $ Number of rows of network outputs

I $\ \ \ \ \ \ \ \ $ Inputs of network

O and $\hat{\boldsymbol{O}}$ $\ \ \ \ \ \ \ \ $ The actual values and output values of all variables

$O$ and $\hat{{O}}$ $\ \ \ \ \ \ \ \ $ The actual values and output values of any single variable

L $\ \ \ \ \ \ \ \ $ Outputs of hidden layer

Loss_phy $\ \ \ \ \ \ \ \ $ Loss standing for physical inconsistency

Loss_mse $\ \ \ \ \ \ \ \ $ Loss standing for mean square error

S_fx and S_fy $\ \ \ \ \ \ \ \ $ Data scale when normalizing milling force

λ_phy $\ \ \ \ \ \ \ \ $ Proportion of physical loss

MSE $\ \ \ \ \ \ \ \ $ The operator for computing mean square error

g(·) $\ \ \ \ \ \ \ \ $ The nonlinear mapping in activation function

l $\ \ \ \ \ \ \ \ $ Autoregressive time step

$\boldsymbol{a}_{1}$ $\ \ \ \ \ \ \ \ $ The first or second coefficients matrices of AR model

$\boldsymbol{a}_{1}, \boldsymbol{a}_{2}, \text { and } \boldsymbol{b}_{0}$ $\ \ \ \ \ \ \ \ $ Coefficients matrices of AR model

Rev $\ \ \ \ \ \ \ \ $ The operator for computing the reciprocal of each element in a matrix

r and γ $\ \ \ \ \ \ \ \ $ Radial and angular runout parameters

R $\ \ \ \ \ \ \ \ $ Cutter radius

e $\ \ \ \ \ \ \ \ $ Error of fitting force under different initial immersion angles and runout parameters

ε(·) $\ \ \ \ \ \ \ \ $ Error of fitting displacement under different vibration parameters

W and B $\ \ \ \ \ \ \ \ $ Weight and bias of neuron

W_n and B_n $\ \ \ \ \ \ \ \ $ Updated values of weight and bias

A $\ \ \ \ \ \ \ \ $ The matrix containing milling force computation information

$F_{x s i} $ and $ F_{y s i}$ $\ \ \ \ \ \ \ \ $ Simulated milling force in X, Y directions

$F_{x m} $ and $ F_{ym}$ $\ \ \ \ \ \ \ \ $ Measured milling force in X, Y directions

$A_{xsi} $ and $ A_{ysi}$ $\ \ \ \ \ \ \ \ $ Simulated acceleration in X, Y directions

$A_{xm} $ and $ A_{ym}$ $\ \ \ \ \ \ \ \ $ Measured acceleration in X, Y directions

a $\ \ \ \ \ \ \ \ $ Index of layer

b $\ \ \ \ \ \ \ \ $ Index of neuron

j $\ \ \ \ \ \ \ \ $ Iterations of updating model parameters

$F_{x} $ and $ F_{y}$ $\ \ \ \ \ \ \ \ $ The milling force in X, Y directions

$F_{xd} $ and $ F_{yd}$ $\ \ \ \ \ \ \ \ $ The dynamic milling force in X, Y directions

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