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

Meiyazhagan Jaganathan, Vikram Pakrashi, Aasifa Rounak

arXiv:2610.06230v1cs.LGnlin.CD

TL;DR

Machining DDEs combine regenerative delay and nonsmooth friction, making accurate physical-parameter estimation important. The paper uses sequential multi-domain XPINNs and reports reduced inverse-estimation errors and low estimation error under noisy data.

  • Problem

    Accurate physical-parameter estimation is important for reliable DDE models, while regenerative delay and friction produce nonsmooth machining dynamics with memory.

    Source: §1 · p. 5

  • Method

    The XPINN framework assigns a network to each successive temporal subdomain, training them sequentially and freezing each completed interval’s parameters.

    Source: §6 · p. 31

  • Results

    I-XPINN reduced Cy error by about 71.5% (stable) and 95.2% (unstable), and K error by approximately 99.6% and 98.3%, respectively, versus I-SPINN.

    Source: §6 · p. 31

  • Takeaways & Limitations

    Under significant measurement noise, I-XPINN maintained low parameter-estimation error, and physics-informed constraints guided reconstruction toward the underlying dynamics.

    Source: §6 · p. 31

    Noise effects were not strictly monotonic: EK generally increased with noise, while Cy error varied non-monotonically, possibly due to noise realizations.

    Source: §5.3 · p. 29

Abstract

from arXiv · show

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.

Acronyms

The acronym list defines the paper’s neural-network and machining terminology, while the non-dimensional notation identifies key model variables, losses, and trainable quantities.

Acronyms

  • DDE means delay differential equation; SPINN and XPINN denote single-domain and eXtended physics-informed neural networks, respectively.I-SPINN and I-XPINN name their inverse formulations.Source: Acronyms · pp. 1–2

Non-dimensional

  • The model notation includes spindle speed N, workpiece rotational delay tw, tool displacement Y(t), and friction coefficients µ, µd, and µs.The listed definitions also include depth of cut ap, damping coefficients c and Cy, and forces Fc, Fd, Ff, and Fr.Source: Acronyms · p. 2
  • Non-dimensional variables include process damping cy, cutting force ff, chip thickness h, relative velocity vγ, depth parameter W, and displacement y.The list also defines states y1 and y2 and the friction-law argument g.Source: Non-dimensional · pp. 2–3
  • PINN losses include initial-condition, data, interface, physical, and shifted-history terms, with adaptive weights for several terms in XPINN.The notation also identifies the fixed random projection matrix Bkd and the residual DDE value R(τi).Source: Non-dimensional · pp. 3–4
  • The notation distinguishes trainable logarithmic parameters Klog and Cy,log from their exponentiated predicted coefficients Kpred and Cy,pred.It also defines reference and noisy solutions, relative errors, and the number of points used for collocation, data, and history losses.Source: Non-dimensional · p. 4

1 Introduction

Machining chatter combines regenerative delay and nonlinear friction, making accurate parameter estimation from system responses challenging. The paper examines PINNs for this setting and develops a multi-domain approach to address limitations involving extended domains, non-constant histories, and noise.

Motivation

  • Regenerative chip-thickness effects depend on the prior workpiece surface, while tool–workpiece friction adds nonlinearities, producing delayed, nonsmooth machining dynamics.These models support analysis of stable and unstable responses and transitions between dynamical states.Source: §1 · p. 5

Research problem

  • Estimating cutting-force and process-damping coefficients is difficult because they depend on operating conditions and are affected by noise, limited observability, nonlinear forces, and delayed dynamics.These coefficients influence cutting forces, machine-tool interactions, system response, and stability predictions.Source: §1 · p. 5

Related work

  • PINN applications in machining had mainly addressed forward prediction and process optimization, while parameter estimation for delayed nonlinear machining dynamics remained unexplored.The introduction positions this work against prior PINN applications to machining and prior PINN methods for DDEs.Source: §1 · p. 5

Research gap

  • Prior PINN-DDE studies addressed forward approximation and inverse identification, but the combined effects of large delays, extended domains, non-constant histories, nonsmoothness, and measurement noise remained insufficiently explored.Large delays also complicate history-dependent representation and accurate propagation of delayed states.Source: §1 · pp. 5–6

Contribution

  • The study evaluates SPINN and develops XPINN with successive temporal subdomains and interconnected networks to improve representation over extended horizons and non-constant histories.The evaluation covers stable and unstable regimes and noise; the authors describe the work as the first XPINN-based parameter identification for nonlinear delayed machining with regenerative and nonsmooth frictional effects.Source: §1 · p. 6

2 Delayed Dynamics of Metal Cutting

The machining model combines regenerative feedback, friction, and process damping in a delayed tool-dynamics system. The study selects cutting-force and process-damping coefficients for inverse estimation and evaluates stable and unstable operating regimes.

2.1 Metal cutting system with regenerative and frictional chatter

  • Turning chatter combines regenerative feedback from the previous workpiece pass with nonlinear friction, so cutting dynamics include both delay and frictional effects.The regenerative contribution depends on current and delayed tool states, while friction depends on instantaneous relative motion.Source: §2.1 · p. 6

2.2 Mathematical formulation of delayed cutting dynamics

  • The model represents the cutting tool as an SDOF mass–spring–damper system, with cutting and feed forces arising from chip–tool interaction.The chip thickness and cutting-force coefficient enter the force model, while friction is represented through the equivalent coefficient µ and Stribeck friction.Source: §2.2 · pp. 7–8

2.3 Unknown parameters and inverse-problem formulation

  • The inverse problem estimates K and Cy while treating the remaining machining parameters as known.K scales cutting force and Cy contributes velocity-dependent process damping; K also appears in the denominator of the non-dimensional damping term, coupling their estimation.Source: §2.3 · p. 9

2.4 Stable and Unstable cases

  • The study evaluates parameter estimation in stable and unstable cutting regimes selected from the stability-lobe diagram.The stable case is ap = 0.4 mm and the unstable chatter case ap = 1.2 mm, both at N = 3600 RPM.Source: §2.4 · p. 10, Figure 3 · p. 10

3 Forward SPINN Formulation for solving machining dynamics

The section formulates a single-domain PINN for the machining DDE and tests its solution accuracy across history functions and temporal horizons. SPINN performs well over a short interval with constant history, but struggles with non-constant histories and extended simulations.

3.1 Single-domain PINN formulation

  • The SPINN represents the DDE solution yθ(τ) with a feedforward ANN and computes solution derivatives using automatic differentiation.The described network uses tanh activation; its trainable weights and biases are optimized by minimizing a loss function.Source: §3.1 · p. 11
  • Its total loss combines a mean-square physics residual over collocation points with an initial-condition loss, while the delayed state comes from the history function before τw and the network afterward.The total loss also uses adaptive weights to normalize individual losses.Source: §3.1 · pp. 11–12

3.2 Data-driven solution of machining system using SPINNs

  • The experiments cover stable cutting at ap = 0.4 mm and unstable or chatter cutting at ap = 1.2 mm, using 5 × 104 collocation points and a 5-hidden-layer network.The network is trained for 2×105 epochs with Adam; the reference solution is computed using MATLAB's DDE23 solver.Source: §3.2 · p. 12
  • Under the constant history function ΦH = 1, SPINN accurately approximates both stable and unstable cases, with relative L2 errors of 7.12×10−3 and 2.6×10−2, respectively.The corresponding MAEs are 1.45×10−3 and 7.54×10−3, respectively.Source: §3.2 · p. 13, Figure 5 · p. 13
  • For ΦH(τ) = sin(τ), SPINN performs poorly in the stable case (L2 9.72 × 10−1; MAE 3.68 × 10−1) and fails in the unstable case (L2 2.76×10−1; MAE 1.38×10−1).Figure 6 compares predicted values with DDE23 references and shows solution scatter for both cases.Source: §3.2 · p. 13, Figure 6 · p. 14
  • Over 0–5τw with constant history, SPINN loses accuracy beyond 2τw, and its predictions fail to capture the machining dynamics.This test uses Nf = 5 × 104 sampling points.Source: §3.2 · p. 13, Figure 7 · p. 15
  • Increasing collocation points from 5×104 to 25×104 and smoothing friction yield only marginal L2 and MAE improvements; SPINN remains unreliable over 5τw.Table 1 compares SPINN error metrics for these settings with constant history.Source: §3.2 · p. 14, Table 1 · p. 15
  • Long-time accuracy matters for determining whether chatter vibrations emerge and persist, while machining history can represent workpiece waviness as a superposition of harmonic functions.The section motivates reliable approximation with non-constant histories and extended simulation horizons.Source: §3.2 · p. 15

4 Forward XPINN formulation for solving machining dynamics

The forward XPINN partitions the machining DDE into delay-length subdomains and trains local subnetworks under physics, history, and interface constraints. Adaptive sampling and sequential training support solution reconstruction over extended intervals with non-constant history.

4.1 Data preparation and adaptive sampling

  • The interval [0,5τw] is split into five equal delay-length subdomains, each with a subnet; residual-based sampling selects the highest-residual collocation points each epoch.Candidate points are generated uniformly within each subdomain before the highest-residual points are retained.Source: §4.1 · p. 17, Figure 11 · p. 18, Figure 10 · p. 18
  • The physical time input is normalized to [−1,1] before training to provide a bounded standardized domain and mitigate scaling issues.The section describes this normalization as typically improving optimization performance.Source: §4.1 · pp. 17–18

4.2 XPINN subnet architecture

  • Each subnet embeds normalized time with fixed random Fourier features, then uses tanh layers and residual blocks with skip connections.The embedding contains 64 sine and 64 cosine components; each subnet covers a local temporal interval.Source: §4.2 · pp. 18–19, Figure 12 · p. 19

4.3 Delay evaluation and total loss

  • Delayed values come from the prescribed history when within its interval and otherwise from the preceding subnet; interface continuity links adjacent subdomains.The delayed-state construction uses only previously available past information.Source: §4.3 · pp. 19–20
  • The loss combines the physics constraint with first-domain shifted-history anchoring and adaptive constraint weights, which help stabilize training across subdomains.Shifted-history anchoring constrains multiple points across the first interval, rather than only the initial point, reducing error propagation to later subdomains.Source: §4.3 · pp. 19–20

4.4 Training Implementation

  • Subnets are trained sequentially: earlier subnetworks are frozen, and each new subnet uses their outputs for delayed states while Adam updates the active parameters.The learning rate decays exponentially, with η0 = 10−3 and γd = 0.99995; the domain-decomposed strategy is described as suited to long-term DDE integration.Source: §4.4 · p. 20

4.5 Data-driven solution of machining system using XPINNs

  • Relative L2 errors were 8.24×10−2 (stable) and 5.15×10−2 (unstable), with MAEs 1.53×10−2 and 2.2×10−2 against DDE23 over five subdomains.The comparison uses a non-constant history function over τ ∈[0,5τw].Source: Figure 13 · p. 21, §4.5 · p. 21

5 Inverse PINN framework for estimating the machining parameters

The inverse framework jointly estimates machining parameters and the system response using DDE physics and measured-response data. Across the reported tests, I-XPINN improves parameter estimation over I-SPINN and retains solution reconstruction accuracy on extended intervals with non-constant histories and noisy data.

5 Inverse PINN framework for estimating the machining parameters

  • The inverse formulation makes unknown physical parameters trainable alongside network weights and biases, adding a data loss to match measured response while retaining DDE consistency.The unknown coefficients are optimized on a logarithmic scale, which keeps estimates positive and reduces their magnitude disparity.Source: §5 · p. 22

5.1 Parameter estimation using I-SPINNs

  • For I-SPINN, process-damping errors reach 26.0348% in the stable case and 53.3091% in the unstable case, with larger solution-reconstruction errors in the unstable case.The reported unstable-case reconstruction errors are L2 = 1.37×10^-1 and MAE = 2.7×10^-2.Source: §5.1 · p. 24

5.2 Parameter estimation using I-XPINNs

  • I-XPINN reduces process-damping relative error by approximately 71.5% in the stable case and 95.2% in the unstable case versus I-SPINN.Table 2 compares parameter-estimation errors; the reported relative errors in K also fall by approximately 99.6% and 98.3%, respectively.Source: Table 2 · p. 25, §5.2 · pp. 25–26
  • I-XPINN reconstructs the DDE solution over the extended time interval with relative L2 errors of 1.73×10−3 in the stable case and 1.33×10−3 in the unstable case.For inverse parameter identification, training took approximately 1.5 hours for SPINN and 10 hours for XPINN.Source: §5.2 · p. 26

5.3 Parameter estimation using I-XPINNs for data with noise

  • With 1% and 5% measurement noise, I-XPINN estimates both parameters in stable and unstable cases, while parameter-error changes are not monotonic with noise intensity.Noise is added to the reference data used in the data loss; the governing DDE residual remains unchanged.Source: §5.3 · pp. 27, 29
  • I-XPINN reconstruction errors against noise-free solutions are consistently lower than errors against noisy observations, but higher noise can noticeably degrade reconstruction, particularly for the stable case.The authors attribute the closer fit to the underlying response to physics-informed constraints rather than reproducing measurement noise.Source: §5.3 · p. 30

6 Conclusion

The conclusion presents multi-domain XPINN as a response to SPINN’s limitations on longer time intervals and non-constant histories. It reports improved forward accuracy, inverse parameter estimation, and noisy-data reconstruction, and frames the approach as a strategy for nonlinear time-delay inverse problems beyond one characteristic delay.

6 Conclusion

  • XPINN reduced relative L2 error by approximately 73.2% in stable and 92.1% in unstable machining, relative to the best-performing SPINN configuration.The conclusion reports that SPINN can approximate delayed dynamics over relatively short horizons but has limitations with non-constant histories and longer intervals.Source: §6 · pp. 30–31
  • I-XPINN reduced Cy error by about 71.5% and 95.2% and K error by approximately 99.6% and 98.3% in stable and unstable cases, respectively, versus I-SPINN.I-XPINN solution reconstruction also achieved relative L2 errors on the order of 10^-3 across both regimes over the extended temporal domain.Source: §6 · p. 31
  • With Gaussian measurement noise, I-XPINN maintained low parameter-estimation error, while its reconstruction errors against the deterministic solution remained lower than against noisy observations.The authors attribute the reconstruction pattern to physics-informed constraints guiding recovery of the underlying dynamics.Source: §6 · p. 31
  • The work presents a systematic strategy for inverse problems in nonlinear time-delay systems over extended horizons exceeding one characteristic delay.The stated challenges include delayed interactions, nonlinear frictional dynamics, and limited parameter observability under measurement noise.Source: §6 · p. 31
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