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Variational Physics-Informed Neural Networks For Solving Partial Differential Equations

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arxiv 1912.00873 v1 pith:RVM3FRWA submitted 2019-11-27 cs.NE cs.LGcs.NAmath.NAphysics.comp-phstat.ML

E. Kharazmi , Z. Zhang , G. E. Karniadakis This is my paper · ORCID
classification cs.NEcs.LGcs.NAmath.NAphysics.comp-phstat.ML
keywords networksneuralvariationalpinnsspacedifferentialphysics-informedtextit
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Physics-informed neural networks (PINNs) [31] use automatic differentiation to solve partial differential equations (PDEs) by penalizing the PDE in the loss function at a random set of points in the domain of interest. Here, we develop a Petrov-Galerkin version of PINNs based on the nonlinear approximation of deep neural networks (DNNs) by selecting the {\em trial space} to be the space of neural networks and the {\em test space} to be the space of Legendre polynomials. We formulate the \textit{variational residual} of the PDE using the DNN approximation by incorporating the variational form of the problem into the loss function of the network and construct a \textit{variational physics-informed neural network} (VPINN). By integrating by parts the integrand in the variational form, we lower the order of the differential operators represented by the neural networks, hence effectively reducing the training cost in VPINNs while increasing their accuracy compared to PINNs that essentially employ delta test functions. For shallow networks with one hidden layer, we analytically obtain explicit forms of the \textit{variational residual}. We demonstrate the performance of the new formulation for several examples that show clear advantages of VPINNs over PINNs in terms of both accuracy and speed.

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Cited by 15 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. OpenAlex reports about 198 citations worldwide. Full citation record

  1. Double screening in the training dynamics of variational physics-informed neural networks for heterogeneous coupled parabolic systems

    math.AP 2026-07 conditional novelty 7.0 of 10

    In the neural tangent kernel regime, the slowest training mode of variational PINNs for heterogeneous parabolic systems is asymptotically determined by the diffusive block alone, with condition number growing like Pe².

  2. HypNO: A Graph-Based Neural Operator with Physics-Informed Message Passing for Hyperbolic Conservation Laws

    cs.LG 2026-07 conditional novelty 7.0 of 10

    A physics-gated space-time graph neural operator reports lower errors than FNO, WENO5, Godunov, and HLL on 1D LWR/ARZ shock benchmarks, backed by a domain-of-dependence receptive-field design rule.

  3. Non-Asymptotic Variational Learning for Monotone Nonlinear Multiscale Elliptic Equations: Scale-Robust Primal-Dual Bounds and Strong-Form Statistical Ill-Conditioning

    math.NA 2026-07 conditional novelty 6.0 of 10

    Variational and primal-dual training of multiscale PDE networks have epsilon-uniform error bounds, while strong-residual training classes provably have Rademacher complexity at least 1/(ε√N) and 1/(ε²√N).

  4. Prediction error certification for PINNs: Theory, computation, and application to Stokes flow

    math.NA 2025-08 conditional novelty 6.0 of 10

    A modified semigroup-based error bound makes PINN error certification applicable to Stokes flow, but the practical certification is not fully rigorous.

  5. A Learning-based Domain Decomposition Method

    cs.LG 2025-07 conditional novelty 6.0 of 10

    A single pretrained neural operator can act as a reusable local solver inside additive Schwarz iterations, solving elliptic PDEs with random microstructures on large, non-convex domains.

  6. A posteriori analysis of neural network approximations

    math.NA 2025-07 conditional novelty 6.0 of 10

    For well-posed variational problems, the error of any approximation, including a neural network, is equivalent to the sum of a computable discrete residual and an estimated remainder.

  7. A discontinuous Galerkin plane wave neural network method for Helmholtz equation and Maxwell's equations

    math.NA 2025-06 conditional novelty 6.0 of 10

    A recursive Galerkin neural network with plane wave activations solves Helmholtz and Maxwell equations with proven convergence, no bounded-parameter assumption, and near-unit condition numbers.

  8. Weak TransNet: A Petrov-Galerkin based neural network method for solving elliptic PDEs

    math.NA 2025-06 conditional novelty 6.0 of 10

    A Petrov-Galerkin neural method with frozen random features as trial space and radial-basis test functions solves elliptic PDEs by least squares, with Fourier and partition-of-unity extensions for multiscale and singu...

  9. Multi-Resolution Training-Enhanced Kolmogorov-Arnold Networks for Multi-Scale PDE Problems

    physics.comp-ph 2025-07 conditional novelty 5.0 of 10

    MR-PIKAN, a multi-resolution training schedule that alternates coarse and fine collocation grids, cuts training time while keeping accuracy on multi-scale forward and inverse PDE problems.

  10. Data-Driven Adaptive Gradient Recovery for Unstructured Finite Volume Computations

    math.NA 2025-07 conditional novelty 5.0 of 10

    A neural-network gradient corrector for unstructured finite volume solvers reports 20-60 percent accuracy gains on 2D Euler benchmarks with improved mesh convergence.

  11. Discontinuity-aware KAN-based physics-informed neural networks

    physics.comp-ph 2025-07 conditional novelty 5.0 of 10

    A discontinuity-aware PINN with adaptive Fourier features, DyT-plus-spline KAN activations, and learned local viscosity captures shocks with errors between 0.9% and 5% on benchmark PDEs and airfoil flows.

  12. Rank Inspired Neural Network for solving linear partial differential equations

    math.NA 2025-06 conditional novelty 5.0 of 10

    Covariance-driven orthogonalization of hidden-layer outputs plus PDE-residual early stopping reduces initialization sensitivity in PIELM for linear PDEs.

  13. PG-KINN: A Physics-Informed Petrov-Galerkin Kolmogorov-Arnold Network for Solving Forward and Inverse PDEs

    cs.LG 2026-07 reject novelty 4.0 of 10

    PG-KINN pairs a KAN trial space with a Petrov–Galerkin test space for forward and inverse PDEs, but the inverse benchmark data is inconsistent with the governing equation and the accuracy claims are not supported by t...

  14. Physics-informed Multiresolution Wavelet Neural Network Method for Solving Partial Differential Equations

    math.NA 2025-08 conditional novelty 4.0 of 10

    A wavelet-basis network trained by least squares solves a range of steady and time-dependent PDEs and captures high-frequency features better than standard PINNs.

  15. PINNs Algorithmic Framework for Simulation of Nonlinear Burgers' Type Models

    cs.LG 2025-06 conditional novelty 4.0 of 10

    Standard physics-informed neural networks reproduce analytic solutions for five coupled Burgers benchmarks with small reported errors, but the paper omits code, training details, and architectures needed to verify or ...

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