Organizations: UCD Centre for Mechanics, Dynamical Systems and Risk Laboratory, School of Mechanical and Materials Engineering, University College Dublin, D04 V1W8, Dublin, Ireland
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 ap=0.4 mm, while the unstable (chatter) cutting condition corresponds to ap=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 ΦH=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 ΦH(τ)=sin(τ) . (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 τ∈[0,5τw] with a constant history function ΦH(τ)=1 and Nf=5×104 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 τ∈[0,5τw] with a constant history function ΦH(τ)=1 and Nf=25×104 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 τ∈[0,5τw] with a constant history function ΦH(τ)=1 and Nf=25×104 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.
Nf
L2
MAE
Stable
Unstable
Stable
Unstable
5×104
3.42×10−1
6.61×10−1
4.82×10−2
2.06×10−1
25×104
3.39×10−1
6.5×10−1
4.78×10−2
2×10−1
25×104 (smooth)
3.08×10−1
6.52×10−1
4.25×10−2
2.01×10−1
Table 1: Comparison of SPINN error metrics for different numbers of collocation points over the interval τ∈[0,5τw] 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 τ∈[0,5τw] with a non-constant history function ΦH(τ)=sin(τ) . (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 Klog and Cy,log of machining system ( 9a ) for simulation time τ∈[0,1.3τw] with a constant history function ΦH(τ)=1 . (a) and (b) represent the values of Klog and Cy,log over the epochs for stable and unstable cases respectively. The blue and red dotted horizontal lines indicate the true values of Klog and Cy,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 Klog and Cy,log for the (a) stable case and (b) unstable case reference data over the simulation interval τ∈[0,5τw] using the history function ΦH(τ)=sin(τ) . The blue and red dotted horizontal lines indicate the true values of Klog and Cy,log , respectively.
Relative Error %
Stable
Unstable
I-SPINN
I-XPINN
I-SPINN
I-XPINN
EK
0.6088%
0.0027%
1.0277%
0.0178%
ECY
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 Klog and Cy,log for the (a) stable case and (b) unstable case with noisy reference data of noise intensity αn=0.01 over the simulation interval τ∈[0,5τw] using the history function ΦH(τ)=sin(τ) . The blue and red dotted horizontal lines indicate the true values of Klog and Cy,log , respectively.
Figure 20: I-XPINN results in estimating the values of Klog and Cy,log for the (a) stable case and (b) unstable case with noisy reference data of noise intensity αn=0.05 over the simulation interval τ∈[0,5τw] using the history function ΦH(τ)=sin(τ) . The blue and red dotted horizontal lines indicate the true values of Klog and Cy,log , respectively.
Region
αn
Kpred[×109]
EK[%]
Cy,pred[×105]
ECy[%]
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 αn .
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 αn used in yrefnoise during training
1%
5%
Stable
Unstable
Stable
Unstable
L2
ypred vs yref
2.61×10−3
5.60×10−3
4.20×10−2
8.56×10−3
ypred vs yrefnoise
9.80×10−3
1.06×10−2
6.29×10−2
4.57×10−2
MAE
ypred vs yref
4.94×10−4
1.89×10−3
5.85×10−3
3.35×10−3
ypred vs yrefnoise
2.11×10−3
4.69×10−3
1.26×10−2
2.07×10−2
Table 4: Relative L2 and MAE values of the I-XPINN reconstructed solution evaluated against both the noise-free DDE23 solution and the measurement noise-added reference solution.
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.
Youyuan Long, Gokhan Solak, Arash Ajoudani
Human-Robot Interfaces and Interaction Lab, Istituto Italiano di Tecnologia, 16163 Genoa, Italy
Accurate dynamics modeling of Brushless DC (BLDC) motors is fundamental to high-performance robotic joint control. This paper presents a Physics-Informed Neural Network (PINN) with a deep residual (ResNet) backbone that learns a continuous-time surrogate of the full six-state BLDC motor dynamics. Given simulation time, applied three-phase voltages, and excitation parameters as inputs, the network directly predicts all motor state variables -- rotor angle, angular velocity, three-phase currents, and winding temperature -- while simultaneously satisfying the governing electromechanical and thermal ODEs through a composite physics-data loss. A curriculum scheduling strategy gradually activates the physics penalty to prevent premature convergence. Training runs are completed in under two minutes on a standard CPU. Crucially, once trained, PINN inference achieves latencies of 0.1--22, mu s per query, up to 118x faster than conventional ODE solvers, making it suitable for real-time observer and control applications.
Haitham El-Hussieny
Department of Mechatronics and Robotics Engineering, Egypt-Japan University of Science and Technology (E-JUST), New Burg El-Arab City, Alexandria, Egypt
Physics-informed neural networks (PINNs) incorporate governing equations into neural-network training and can approximate PDE solutions without requiring large observational datasets. Parameterized PINNs (ParamPINNs) further take physical parameters as inputs, allowing a single model to represent a family of PDE solutions over a parameter domain. Existing ParamPINNs, however, still face inefficient training, uneven accuracy across parameters, and overfitting to a limited set of sampled parameter tasks, which can impair generalization to unsampled parameters. To address these issues, we propose a continual-learning physics-informed neural network (CL-PINN), which treats PDE instances at different parameter values as related tasks and learns them sequentially. CL-PINN combines Bayesian-optimization-based active parameter selection, task-wise dynamic loss weighting, sparse physics-constrained replay, and an optional parameter subnetwork to improve task allocation and knowledge retention under bounded active-task capacity. It requires no observational data and is designed to solve parameterized PDEs over relatively broad parameter domains under limited computational resources. Multi-seed evaluations on five benchmarks, including one continuous function and four parameterized PDEs, show that Bayesian selection substantially reduces objective-loss queries relative to grid-greedy search, while sparse replay mitigates forgetting of earlier tasks. Under the prescribed within-case resource protocols, CL-PINN generally provides higher and more balanced solution accuracy than fixed-sampling and grid-greedy baselines. CL-PINN offers a practical route toward learning PDE solutions that generalize across physical parameters and has the potential to support reusable physics-informed surrogates for large-scale engineering parameter studies.
Xujia Chen, Xinyue Hu, Letian Chen +2
Department of Automation, Tsinghua University, Beijing, 100084, China