Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2610.03575

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

Pelin G. Geredeli·Mohammad Mahabubur Rahman·Deepanshu Verma
View original source ↗Read source document ↗Export metadata ↓

Abstract

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.

Explore related subjects

math.NAcs.NA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Pelin G. Geredeli, Mohammad Mahabubur Rahman, Deepanshu Verma. 2026-10-02. Error Estimates in Physics-Informed Neural Networks to Heat-Wave Fluid-Structure Interactive System. https://arxiv.org/abs/2610.03575

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A theory of meta-factorization

Meta-factorization asks how different factors and transforms describe the same matrix and how a factor can be replaced while preserving that matrix. Projector and reconstruction equations give a common description. For $A=FGH^*$ with outer factors of full column rank $k$, $Y^*F=H^*X=I_k$ certifies generally oblique projectors. Exact reconstruction requires their ranges to contain the column spaces of $A$ and $A^*$. Two design matrices parametrize every certifying pair; the core is unique and nonsingular at $k=\operatorname{rank}(A)$. Detailed constructions recover SVD, column-pivoted QR, UTV, generalized Nystrom, CUR and classical pseudoinverse formulas with their structural hypotheses. The framework makes factor replacement testable from stored factors. For a supplied candidate $\check F$ and transform $\check Y^*$, with $H$ of full column rank, $\check G=(\check Y^*F)G$ preserves $A$ exactly when $[F-\check F(\check Y^*F)]G=0$. Test, repair and exact factored residual require no access to $A$. With the pseudoinverse as transform, the updated product is a closest reconstruction within the candidate's column space in every unitarily invariant norm. We characterize the accepted subspaces under the stated inverse and rank conditions and extend replacement to different stored widths. Nonsingular generalized right-hand sides change certifying coordinates; finite-order choices yield potent operators and periodic reconstruction. Classical Fourier components test preservation of the core at every step, including singular cores. The contribution is a common description with explicit replacement and reconstruction consequences. Numerical acceptance, conditioning and candidate selection require further analysis.

math.NA↗
arXiv2026

Stochastic Quadrature Rules for Solving PDEs using Neural Networks

We examine the challenges associated with numerical integration when applying Neural Networks to solve Partial Differential Equations (PDEs). We specifically investigate the Deep Ritz Method (DRM), chosen for its practical applicability and known sensitivity to integration inaccuracies. Our research demonstrates that both standard deterministic integration techniques and biased stochastic quadrature methods can lead to incorrect solutions. In contrast, employing high-order, unbiased stochastic quadrature rules defined on integration meshes in low dimensions is shown to significantly enhance convergence rates at a comparable computational expense with respect to low-order methods like Monte Carlo. Additionally, we introduce novel stochastic quadrature approaches designed for triangular and tetrahedral mesh elements, offering increased adaptability for handling complex geometric domains. We highlight that the variance inherent in the stochastic gradient acts as a bottleneck for convergence. Furthermore, we observe that for gradient-based optimisation, the crucial factor is the accurate integration of the gradient, rather than just minimizing the quadrature error of the loss function itself.

math.NA↗
arXiv2026

A Stochastic Genetic Interacting Particle Method for Reaction-Diffusion-Advection Equations

We develop and analyze a stochastic genetic interacting particle method (SGIP) for reaction-diffusion-advection (RDA) equations. The SGIP method employs operator splitting to approximate the advection-diffusion and reaction processes, treating the former using particle drift-diffusion and the latter via exact or implicit integration of reaction dynamics over bins, where particle density is estimated using a histogram. A key innovation is the incorporation of adaptive resampling to close the loop of particle and density field description of solutions, mimicking the selection mechanism in genetics. Resampling is also crucial for maintaining long-term stability by redistributing particles in accordance with the evolving density field. We provide a comprehensive error analysis and establish convergence bounds under appropriate regularity assumptions. Numerical experiments in one to three space dimensions demonstrate the method's effectiveness across various reaction types (Fisher-Kolmogorov-Petrovsky-Piskunov (FKPP), cubic, Arrhenius) and flow configurations (shear, cellular, cat's eye, Arnold-Beltrami-Childress (ABC) flows), showing excellent agreement with the finite difference method (FDM) while offering computational advantages for complex flow geometries and higher-dimensional problems.

math.NA↗