Parameter Estimation in Machining Dynamics with Regenerative Delay and Nonsmooth Friction using Physics-Informed Neural Networks

Meiyazhagan Jaganathan; Vikram Pakrashi; Aasifa Rounak

cs.LGOct 5, 2026

Parameter Estimation in Machining Dynamics with Regenerative Delay and Nonsmooth Friction using Physics-Informed Neural Networks

Authors: Meiyazhagan Jaganathan, Vikram Pakrashi, Aasifa Rounak

Organizations: UCD Centre for Mechanics, Dynamical Systems and Risk Laboratory, School of Mechanical and Materials Engineering, University College Dublin, D04 V1W8, Dublin, Ireland

Abstract

A multi-domain eXtended Physics-Informed Neural Network (XPINN) framework is developed for nonsmooth Delay Differential Equations (DDEs). This is the first implementation to demonstrate the efficacy of partitioning the temporal domain into subdomains of integer multiples of the characteristic time delay and progressively training the associated subnetworks while freezing previously learned parameters. The efficacy of the proposed framework is demonstrated using a machining dynamics model that incorporates both regenerative and nonsmooth frictional effects. Results demonstrate that the proposed multi-domain XPINN framework leads to better solution reconstruction in DDEs and improved parameter estimation compared to a generic PINN (SPINN) formulation. The proposed method works particularly well for extended temporal domains and non-constant history functions. The robustness of inverse XPINN (I-XPINN) is also assessed using reference data contaminated with Gaussian measurement noise. Results indicate that I-XPINN remains resilient to measurement noise and the physics-informed constraints guide the network toward accurately recovering the underlying dynamics. This demonstrates, for the first time, the potential of the proposed framework for reliable parameter identification in DDEs characterised by nonsmoothness and large time delays.

Figures & tables

Figure 1: Schematic representation of the turning process, illustrating the relative positioning and motion of the cutting tool and workpiece.

Figure 2: Illustration of the dynamic model used to describe the turning process, showing the cutting tool as an equivalent mass-spring-damper system subjected to the forces generated during the cutting process.

Figure 3: Stability lobe diagram showing the stable (shaded in purple) and unstable (shaded in green) operating regimes, with the selected parameters for the stable and unstable cutting cases indicated by a star and a cross, respectively. The stable cutting condition corresponds to a depth of cut of $a_{p}=0.4$ mm, while the unstable (chatter) cutting condition corresponds to $a_{p}=1.2$ mm, both at a spindle speed of $N=3600$ RPM.

Figure 4: Schematic of the PINN for solving the forward problem

Figure 5: SPINN performance in solving the machining system with a constant history function $\Phi_{H}=1$ . (a) and (b) represent plots of predicted values (dotted line) over the DDE23 reference solution (solid line) for the stable and unstable cases, respectively. (c) and (d) correspond to the scatter plots of the DDE solution with the PINN solution, respectively, for the stable and unstable cases.

Figure 6: SPINN performance in solving the machining system with a history function $\Phi_{H}(\tau)=\sin(\tau)$ . (a) and (b) represent plots of predicted values (dotted line) over the DDE23 reference solution (solid line) for the stable and unstable cases, respectively. (c) and (d) correspond to the scatter plots of the DDE solution with the PINN solution, respectively, for the stable and unstable cases.

Figure 7: SPINN results in solving the machining system for simulation time $\tau\in[0,5\tau_{w}]$ with a constant history function $\Phi_{H}(\tau)=1$ and $N_{f}=5\times 10^{4}$ sampling points. (a) and (b) represent plots of predicted values (dotted line) over the DDE23 reference solution (solid line) for the stable and unstable cases, respectively. (c) and (d) correspond to the scatter plots of DDE solution with PINN solution, respectively, for stable and unstable cases.

Figure 8: SPINN results in solving the machining system for simulation time $\tau\in[0,5\tau_{w}]$ with a constant history function $\Phi_{H}(\tau)=1$ and $N_{f}=25\times 10^{4}$ sampling points. (a) and (b) represent plots of predicted values (dotted line) over the DDE23 reference solution (solid line) for the stable and unstable cases, respectively. (c) and (d) correspond to the scatter plots of the DDE solution with the PINN solution, respectively, for the stable and unstable cases.

Figure 9: SPINN results in solving the smooth approximated machining system for simulation time $\tau\in[0,5\tau_{w}]$ with a constant history function $\Phi_{H}(\tau)=1$ and $N_{f}=25\times 10^{4}$ sampling points. (a) and (b) represent plots of predicted values (dotted line) over the DDE23 reference solution (solid line) for the stable and unstable cases, respectively. (c) and (d) correspond to the scatter plots of the DDE solution with the PINN solution, respectively, for the stable and unstable cases.

$N_{f}$	$L_{2}$		MAE
$N_{f}$	Stable	Unstable	Stable	Unstable
$5\times 10^{4}$	$3.42\times 10^{-1}$	$6.61\times 10^{-1}$	$4.82\times 10^{-2}$	$2.06\times 10^{-1}$
$25\times 10^{4}$	$3.39\times 10^{-1}$	$6.5\times 10^{-1}$	$4.78\times 10^{-2}$	$2\times 10^{-1}$
$25\times 10^{4}$ (smooth)	$3.08\times 10^{-1}$	$6.52\times 10^{-1}$	$4.25\times 10^{-2}$	$2.01\times 10^{-1}$

Table 1: Comparison of SPINN error metrics for different numbers of collocation points over the interval

\tau\in[0,5\tau_{w}]

and a smooth approximation of the nonsmooth friction with a constant history function.

Figure 10: Schematic of the XPINN

Figure 11: Schematic of residual-based adaptive sampling in XPINN at one epoch

Figure 12: Network architecture of the subnet in XPINN

Figure 13: XPINN results in solving the machining system for simulation time $\tau\in[0,5\tau_{w}]$ with a non-constant history function $\Phi_{H}(\tau)=\sin(\tau)$ . (a) and (b) represent plots of predicted values (dotted line) over the DDE23 reference solution (solid line) for the stable and unstable cases, respectively. (c) and (d) correspond to the scatter plots of the DDE solution with the PINN solution, respectively, for the stable and unstable cases.

Figure 14: Schematic of the inverse PINN for estimating the unknown parameters of the considered machining system.

Figure 15: I-SPINN results in estimating the values of $K_{\log}$ and $C_{y,\log}$ of machining system ( 9a ) for simulation time $\tau\in[0,1.3\tau_{w}]$ with a constant history function $\Phi_{H}(\tau)=1$ . (a) and (b) represent the values of $K_{\log}$ and $C_{y,\log}$ over the epochs for stable and unstable cases respectively. The blue and red dotted horizontal lines indicate the true values of $K_{\log}$ and $C_{y,\log}$ , respectively.

Figure 16: Comparison of the approximated solution with the reference solution of the considered DDE for (a) the stable case and (b) the unstable case with the I-SPINN framework. (c) and (d) correspond to the scatter plots of the DDE solution with the I-SPINN solution, respectively, for the stable and unstable cases.

Figure 17: I-XPINN results in estimating the values of $K_{\log}$ and $C_{y,\log}$ for the (a) stable case and (b) unstable case reference data over the simulation interval $\tau\in[0,5\tau_{w}]$ using the history function $\Phi_{H}(\tau)=\sin(\tau)$ . The blue and red dotted horizontal lines indicate the true values of $K_{\log}$ and $C_{y,\log}$ , respectively.

Relative Error $\%$	Stable		Unstable
Relative Error $\%$	I-SPINN	I-XPINN	I-SPINN	I-XPINN
$E_{K}$	$0.6088\%$	$0.0027\%$	$1.0277\%$	$0.0178\%$
$E_{C_{Y}}$	$26.0348\%$	$7.4179\%$	$53.3091\%$	$2.5507\%$

Table 2: Comparison of relative errors of parameter estimations with I-SPINN and I-XPINN.

Figure 18: Comparison of the approximated solution with the reference solution of the considered DDE for (a) the stable case and (b) the unstable case with the I-XPINN framework. (c) and (d) correspond to the scatter plots of the DDE solution with the I-XPINN solution, respectively, for the stable and unstable cases.

Figure 19: I-XPINN results in estimating the values of $K_{\log}$ and $C_{y,\log}$ for the (a) stable case and (b) unstable case with noisy reference data of noise intensity $\alpha_{n}=0.01$ over the simulation interval $\tau\in[0,5\tau_{w}]$ using the history function $\Phi_{H}(\tau)=\sin(\tau)$ . The blue and red dotted horizontal lines indicate the true values of $K_{\log}$ and $C_{y,\log}$ , respectively.

Figure 20: I-XPINN results in estimating the values of $K_{\log}$ and $C_{y,\log}$ for the (a) stable case and (b) unstable case with noisy reference data of noise intensity $\alpha_{n}=0.05$ over the simulation interval $\tau\in[0,5\tau_{w}]$ using the history function $\Phi_{H}(\tau)=\sin(\tau)$ . The blue and red dotted horizontal lines indicate the true values of $K_{\log}$ and $C_{y,\log}$ , respectively.

Region	$\alpha_{n}$	$K_{\mathrm{pred}}\;[\times 10^{9}]$	$E_{K}\;\mathrm{[\%]}$	$C_{y,\mathrm{pred}}\;[\times 10^{5}]$	$E_{C_{y}}\;\mathrm{[\%]}$
Stable	$0\%$	$6.02017$	$0.0027$	$6.56324$	$7.4179$
	$1\%$	$6.03042$	$0.1731$	$6.86907$	$12.4235$
	$5\%$	$5.95266$	$1.1186$	$6.05781$	$0.8541$
Unstable	$0\%$	$6.01893$	$0.0178$	$6.26585$	$2.5507$
	$1\%$	$6.03175$	$0.1952$	$6.23803$	$2.0955$
	$5\%$	$6.05486$	$0.579$	$6.18253$	$1.1871$

Table 3: Comparison of relative errors of parameter estimations with I-XPINN for various noise intensities

\alpha_{n}

Figure 21: Scattered plots of the I-XPINN approximated solution against the reference DDE23 solutions for the stable and unstable cases. The top and bottom rows correspond to the stable and unstable cases, respectively. Figs. (a)-(d) show the results for the I-XPINN trained with 1% noisy reference data, while Figs. (e)-(h) show the results for the model trained with 5% noisy reference data. The green and red scattered points represent the noise-free DDE23 solution and the noisy reference data, respectively, and the I-XPINN-approximated solution is compared against both datasets.

Error metric	Test data	Value of $\alpha_{n}$ used in $y_{\mathrm{ref}}^{\mathrm{noise}}$ during training
		$1\%$		$5\%$
		Stable	Unstable	Stable	Unstable
$L_{2}$	$y_{\mathrm{pred}}$ vs $y_{\mathrm{ref}}$	$2.61\times 10^{-3}$	$5.60\times 10^{-3}$	$4.20\times 10^{-2}$	$8.56\times 10^{-3}$
$L_{2}$	$y_{\mathrm{pred}}$ vs $y_{\mathrm{ref}}^{\mathrm{noise}}$	$9.80\times 10^{-3}$	$1.06\times 10^{-2}$	$6.29\times 10^{-2}$	$4.57\times 10^{-2}$
MAE	$y_{\mathrm{pred}}$ vs $y_{\mathrm{ref}}$	$4.94\times 10^{-4}$	$1.89\times 10^{-3}$	$5.85\times 10^{-3}$	$3.35\times 10^{-3}$
MAE	$y_{\mathrm{pred}}$ vs $y_{\mathrm{ref}}^{\mathrm{noise}}$	$2.11\times 10^{-3}$	$4.69\times 10^{-3}$	$1.26\times 10^{-2}$	$2.07\times 10^{-2}$

Table 4: Relative

L_{2}

and

MAE

values of the I-XPINN reconstructed solution evaluated against both the noise-free DDE23 solution and the measurement noise-added reference solution.

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Apr 20, 2026, cs.LGJ/K move · Enter open · S save

Dissipative Latent Residual Physics-Informed Neural Networks for Modeling and Identification of Electromechanical Systems

Youyuan Long, Gokhan Solak, Arash Ajoudani

Human-Robot Interfaces and Interaction Lab, Istituto Italiano di Tecnologia, 16163 Genoa, Italy

Parametric Physics-Informed Neural Network Latent Dynamics Residual Learning Chaos Friction First-Principles Calculation

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Accurate dynamical modeling is essential for simulation and control of embodied systems, yet first-principles models of electromechanical systems often fail to capture complex dissipative effects such as joint friction, stray losses, and structural damping. While residual-learning physics-informed neural networks (PINNs) can effectively augment imperfect first-principles models with data-driven components, the residual terms are typically implemented as unconstrained multilayer perceptrons (MLPs), which may inadvertently inject artificial energy into the system. To more faithfully model the dissipative dynamics, we propose DiLaR-PINN, a dissipative latent residual PINN designed to learn unmodeled dissipative effects in a physically consistent manner. Structurally, the residual network operates only on unmeasurable (latent) state components and is parameterized in a skew-dissipative form that guarantees non-increasing energy for any choice of network parameters. To enable stable and data-efficient training under partial measurability of the state, we further develop a recurrent rollout scheme with a curriculum-based sequence length extension strategy. We validate DiLaR-PINN on a real-world helicopter system and compare it against four baselines: a pure physical model (without a residual network), an unstructured residual MLP, a DiLaR variant with a soft dissipativity constraint, and a black-box LSTM. The results demonstrate that DiLaR-PINN more accurately captures dissipative effects and achieves superior long-horizon extrapolation performance.

Abstract

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