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Error Estimates in Physics-Informed Neural Networks to Heat-Wave Fluid-Structure Interactive System

Pelin G. Geredeli, Mohammad Mahabubur Rahman, Deepanshu Verma

arXiv:2610.03575v1math.NAcs.NA

TL;DR

This paper addresses the lack of rigorous error estimates for PINNs approximating coupled fluid-structure interaction PDEs. It analyzes a canonical heat-wave model, proving error bounds for the PINN approximation and relating generalization error to training error and quadrature points.

  • Problem

    Rigorous PINN error estimates for coupled fluid-structure interaction systems, especially parabolic-hyperbolic systems coupled through interface conditions, are lacking.

    Source: §1 · p. 2

  • Method

    The paper analyzes a PINN that minimizes residual losses for the coupled PDEs and their interface and initial conditions, deriving error bounds for the heat-wave system.

    Source: §1 · p. 2, §3.2 · pp. 7, 13

  • Results

    The analysis bounds exact-to-PINN error through residuals and shows that the difference between squared generalization and training errors is controlled by quadrature points.

    Source: §3.2 · pp. 7, 13

  • Takeaways & Limitations

    The work provides the first rigorous error analysis for PINNs applied to a coupled fluid-structure interaction PDE system, in the canonical heat-wave setting.

    Source: §1 · p. 2

    The analysis uses zero source functions and requires C4 regularity of the heat and structure solutions for the stated error estimates.

    Source: §2 · p. 3, §3.2 · p. 13

Abstract

from arXiv · show

We study a physics-informed neural network (PINN) approximation of a coupled PDE system that models the interaction between fluid and elastic dynamics. The canonical heat-wave model considered here couples a parabolic and a hyperbolic equation through transmission conditions on a fixed boundary interface, and serves as a prototype for the Stokes-elastic systems that arises in real world applications. This coupling makes the error analysis more delicate than that for a single PDE. We bound the error between the exact and PINN solutions by the associated PINN residuals, and we moreover prove that the generalization error is controlled by the training error, the number of quadrature points, and the size of the network. {To the best of our knowledge}, these are the first error estimates for a PINN approximation of a coupled fluid-structure interaction PDE system. We also show that numerical experiments with respect to a given test problem.

1. Introduction

Rigorous PINN error analysis for coupled fluid-structure systems remains limited, particularly for parabolic-hyperbolic systems coupled through interface conditions. This paper analyzes a canonical heat-wave model and addresses whether small residual-based errors imply small solution error.

1. Introduction

  • Earlier PINN error analyses covered standard and several nonlinear PDEs, but rigorous justification for coupled multiphysics systems remained limited.The cited coupled Navier-Stokes and elastic-disc PINN study compared against a classical solver without error analysis; the cited literature lacked estimates for parabolic-hyperbolic systems coupled through interface conditions.Source: §1 · p. 2
  • The paper develops the first rigorous error analysis for PINNs approximating a coupled fluid-structure interaction system, addressing a gap even in the canonical case.Prior work applied PINNs to a fluid-structure problem without error analysis, while the parabolic-hyperbolic interface setting lacked results guaranteeing that small training loss implies small solution error.Source: §1 · p. 2
  • The analysis asks whether small generalization error ensures small total solution error and whether small training error with enough samples yields proportionately small generalization error.These questions concern the exact solution and its PINN approximation, with total error measured in a relevant energy norm.Source: §1 · p. 2
  • The PINN minimizes residual losses for the PDEs, interface, and initial conditions at collocation points in a mesh-free formulation.The paper derives error bounds and reports numerical experiments for a model coupling parabolic heat and hyperbolic wave dynamics at a fixed interface; analysis of the more complex Navier-Stokes and elastic wave/plate model is deferred.Source: §1 · p. 2

1.1. Description of the Coupled Heat-Wave Interaction PDE Problem

  • The elastic structure domain Ωs is completely embedded within the bounded heat domain Ωf (d = 2 or 3), and heat–structure dynamics couple along the fixed interface Γs = ∂Ωs.The model uses u for fluid flow, w for structure displacement, and wt for structure velocity.Source: §1.1 · p. 3

2. Preliminaries

The preliminaries frame the heat-wave system as a simplified counterpart of a coupled Stokes-fluid and elastic-structure model. They set up a semigroup formulation for wellposedness and invoke a higher-regularity result from a related Stokes-wave system.

2. Preliminaries

  • For the theoretical error analysis, the source functions are set to zero, and the heat equation replaces the Stokes equation in the simplified coupled model.The model is presented as a canonical case related to the coupled 3D Stokes-fluid and 3D Lamé-elastic system in.Source: §2 · p. 3
  • The coupled system is formulated as an abstract ODE in a Hilbert space, and its generator defines a strongly continuous contraction semigroup that yields a unique solution for any initial data in H.For initial data in D(A), Theorem 2.1 also states additional solution regularity.Source: §2 · p. 4
  • The error analysis relies on a higher-regularity result for a Stokes-wave interaction system, assuming initial data in D(Ã) and w0 in H2(Ωs).The cited result establishes higher regularity with continuous dependence on the initial data [4, Theorem 2.1].Source: §2 · p. 4

3. Neural Networks for the PDE system (1)

This section develops a shared-network PINN framework for the coupled heat-wave system, defining residual and quadrature quantities and deriving error estimates. It bounds the trained PINN’s L2 error in terms of training error and quadrature-point counts, under higher solution regularity with constants bounded using network parameters and the activation function.

3. Neural Networks for the PDE system (1)

  • A single shared network maps (t, x) to 2d outputs split into fluid heat uθ and solid wave wθ components, evaluated on their respective domains.The activation is required to be C4 so fourth-order derivatives in the PDE residuals are classically defined.Source: §3 · p. 5

3.1. Quadrature rules

  • Midpoint quadrature replaces residual integrals with computable sums, but its accuracy near curved-domain boundaries drops to first order when the Cartesian grid is not aligned.The stated quadrature-error condition assumes g ∈ C2(Λ).Source: §3.1 · pp. 5–6

3.2. Physics-Informed Neural Networks

  • Theorem 3.3 derives an energy estimate for the L2 error between the exact coupled heat-wave solution and its PINN approximation for d ∈ {2, 3}.The system residuals cover the fluid and solid interiors, outer boundary, interface, and initial conditions; they vanish for the exact solution.Source: §3.2 · pp. 6–7
  • Generalization error measures residuals over the full domains, while training error uses finite weighted quadrature samples; midpoint estimates relate them for sufficiently large M.The trained PINN is determined by minimizing the training error, which is decomposed into PDE, boundary, interface, and initial errors.Source: §3.2 · pp. 12–13
  • The trained PINN’s L2 error is bounded using its training error and the numbers of quadrature points.The quadrature counts correspond to initial, PDE, outer-boundary, and interface quadratures.Source: §3.2 · p. 13
  • Theorem 3.7 requires u and w to have C4 regularity on their respective space-time domains.The paper notes that sufficiently smooth initial data and bootstrapping yield higher Sobolev regularity for the coupled system’s solution.Source: §3.2 · pp. 13–14
  • The proof bounds each residual using training error and quadrature counts, then estimates the resulting constants from network parameters L, W, R and activation function σ.It proceeds in three parts: residual bounds, bounds on constants, and combination of those estimates into the theorem.Source: §3.2 · pp. 14, 16, 17

4. Numerical Experiment

The numerical experiments test the heat-wave FSI error estimates by varying residual quadrature points and network width. They report that increasing quadrature points reduces both training and L2 errors, while L2 error changes little with width at fixed quadrature count.

4. Numerical Experiment

  • The manufactured test uses a circle-in-circle geometry with Rs = 0.5 and Rf = 1, and exact fields w1 = e^tϕ(r) + 1 and w2 = 2e^tϕ(r) + 2.Here ϕ(r) = 1 − r^2.Source: §4 · p. 17
  • A single fully connected network maps (x, y, t) to the coupled fields, using two hidden layers of width W with tanh activation; automatic differentiation supplies wt.The network output dimension is four.Source: §4 · p. 18
  • Figure 3 compares manufactured fields, PINN approximations, and pointwise absolute error at t = 1 for M = 21,200 and W = 80.Its caption identifies u and w on the top row and u and wt on the bottom row.Source: Figure 3 · p. 18, §4 · p. 19
  • Training uses Adam for 20,000 epochs, followed by six L-BFGS cycles of 1,000 iterations each to further reduce PDE, boundary, and interface residuals.Adam uses an initial learning rate of 5 × 10^-4, reduced on plateaus, with gradient clipping throughout.Source: §4 · p. 19
  • Increasing M reduces training loss, total L2-error, and the theoretical bound, with the largest reported reduction from 21,200 to 35,200 points.The experiments are presented as validation of Theorem 3.7; the quadrature-error terms in (81) decrease as M increases.Source: §4 · pp. 17, 19, Figure 2 · p. 18
  • At fixed M = 35,200, L2-error varies only slightly across widths 20, 80, and 160, while training loss falls from W = 20 to W = 80 and then changes little.The measured error remains below the plotted bound in every run; the bound's width-dependent constants are worst-case and need not be sharp for a trained network.Source: Figure 2 · p. 18, §4 · p. 19

5. Appendix

The appendix introduces technical lemmas used to bound neural-network derivatives. It also outlines a proof step for a C^n function estimate.

5. Appendix

  • Lemma 5.1 assumes g ∈ C^n(D), with D ⊂ R^d and d = 2 or 3.Source: §5 · p. 19
  • The proof of Lemma 5.1 uses the L∞(D)-norm, the triangle inequality, and the C^n(D)-norm, then takes a maximum over multi-indices with |α| ≤ n.Source: §5 · pp. 19–20
  • Lemma 5.2 is introduced to obtain necessary bounds on neural-network derivatives and assumes C^n regularity for its functions.Source: §5 · p. 20

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