REVIEW 9 cited by
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
On the convergence of physics informed neural networks for linear second-order elliptic and parabolic type PDEs
read the original abstract
Physics informed neural networks (PINNs) are deep learning based techniques for solving partial differential equations (PDEs) encounted in computational science and engineering. Guided by data and physical laws, PINNs find a neural network that approximates the solution to a system of PDEs. Such a neural network is obtained by minimizing a loss function in which any prior knowledge of PDEs and data are encoded. Despite its remarkable empirical success in one, two or three dimensional problems, there is little theoretical justification for PINNs. As the number of data grows, PINNs generate a sequence of minimizers which correspond to a sequence of neural networks. We want to answer the question: Does the sequence of minimizers converge to the solution to the PDE? We consider two classes of PDEs: linear second-order elliptic and parabolic. By adapting the Schauder approach and the maximum principle, we show that the sequence of minimizers strongly converges to the PDE solution in $C^0$. Furthermore, we show that if each minimizer satisfies the initial/boundary conditions, the convergence mode becomes $H^1$. Computational examples are provided to illustrate our theoretical findings. To the best of our knowledge, this is the first theoretical work that shows the consistency of PINNs.
Forward citations
Cited by 9 Pith papers
-
Universal Approximation of Nonlinear Operators and Their Derivatives
Proves the first universal approximation theorems for k-times differentiable nonlinear operators between Banach spaces and their derivatives uniformly on compact sets in weighted Sobolev norms via encoder-decoder oper...
-
Parameterized Representations via Implicit Stochastic Modulation for High-Dimensional and High-Order Neural PDE Solvers
PRISM enables zero-shot parameterized high-dimensional high-order neural PDE solvers via implicit stochastic modulation that decouples parameters from the differentiation graph while preserving unbiased estimators.
-
Effective Dimensionality as an Operator Invariant for Physics-Preserving Constraint Adaptation in Physics-Informed Neural Networks
d_eff in PINNs is shown to be an operator invariant equal to kernel dimension for finite-kernel operators, enabling subspace projection for physics-preserving constraint adaptation.
-
Universal Approximation of Nonlinear Operators and Their Derivatives
Proves first UATs for k-times differentiable nonlinear operators and their derivatives via OL architectures uniformly on compact sets in weighted Bastiani-Sobolev spaces on general Banach spaces.
-
Uncertainty-aware damage identification in short-span bridges via physics-informed variational autoencoder
A PI-GCVAE with a differentiable eigenvalue decoder and Gaussian-copula latents recovers true stiffness posteriors on noisy synthetic short-span bridge data at ~79% 95%-coverage.
-
XNet-Enhanced Deep BSDE Method and Numerical Analysis
Establishes convergence for non-Lipschitz generators via bounded double-well lemma and truncated BSDE analysis, plus XNet architecture for efficient 100D PDE computation.
-
Error Analysis of Tr-PINNs Algorithm for 2D Incompressible Navier-Stokes Equations with Non-Homogeneous Boundary Conditions
Tr-PINNs corrects boundary errors in PINNs for non-homogeneous 2D Navier-Stokes equations and supplies error analysis derived from the non-homogeneous Stokes problem.
-
PINNsur: Physics-Informed Neural Networks for PDEs on Curved Surfaces
PINNSur applies PINNs to surface PDEs by neural approximation of normals and operator projection, with an added empirical test for convergence behavior.
-
A Practitioner's Guide to Kolmogorov-Arnold Networks
A systematic review of Kolmogorov-Arnold Networks that maps their relation to Kolmogorov superposition theory, MLPs, and kernels, examines basis-function design choices, summarizes performance advances, and supplies a...
discussion (0)
Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.