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Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling
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Despite the success of physics-informed neural networks (PINNs) in approximating partial differential equations (PDEs), PINNs can sometimes fail to converge to the correct solution in problems involving complicated PDEs. This is reflected in several recent studies on characterizing the "failure modes" of PINNs, although a thorough understanding of the connection between PINN failure modes and sampling strategies is missing. In this paper, we provide a novel perspective of failure modes of PINNs by hypothesizing that training PINNs relies on successful "propagation" of solution from initial and/or boundary condition points to interior points. We show that PINNs with poor sampling strategies can get stuck at trivial solutions if there are propagation failures, characterized by highly imbalanced PDE residual fields. To mitigate propagation failures, we propose a novel Retain-Resample-Release sampling (R3) algorithm that can incrementally accumulate collocation points in regions of high PDE residuals with little to no computational overhead. We provide an extension of R3 sampling to respect the principle of causality while solving time-dependent PDEs. We theoretically analyze the behavior of R3 sampling and empirically demonstrate its efficacy and efficiency in comparison with baselines on a variety of PDE problems.
Forward citations
Cited by 9 Pith papers
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EvoPINN: Agentic Discovery of Executable Algorithms for Physics-Informed Neural Networks
An LLM-guided, execution-verified evolutionary search discovered PINN training algorithms that beat the seed network on four PDE benchmarks and matched expert-designed baselines on three.
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RAMS: Residual-based adversarial-gradient moving sample method for scientific machine learning in solving partial differential equations
Treating training samples as trainable parameters and moving them along the residual's adversarial gradient improves accuracy across PINN and operator learning benchmarks.
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Energy Dissipation Rate Guided Adaptive Sampling for Physics-Informed Neural Networks: Resolving Surface-Bulk Dynamics in Allen-Cahn Systems
An energy-dissipation-rate-guided sampling strategy (EDRAS) improves PINN accuracy on thermodynamically consistent Allen-Cahn models, giving up to a sixfold relative MSE reduction over residual-based adaptive refineme...
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Energy Manifold Natural Gradient Descent: Riemannian Optimization for Neural PDE Solvers
Energy Manifold Natural Gradient Descent (EMNGD) defines the energy natural gradient on a Riemannian parameter manifold and proves it equals the energy-metric projection of the function-space Newton step.
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Trainable Spline Representations for Physics-Informed Learning
A tensor-product B-spline whose coefficients are learned by minimizing PDE residuals achieves lower error than standard PINNs on four benchmark problems with far fewer parameters.
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LIGO-PINN: Learned Initialization via Gated Optimization to Alleviate Convergence Failures in Physics Informed Neural Networks
Meta-learning on easy PDE tasks plus a layer-wise gating schedule reduces extrapolation error by about 91% relative to six PINN baselines on hard convection, Helmholtz, and Navier-Stokes benchmarks.
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I-FENN with DeepONets: accelerating simulations in coupled multiphysics problems
A hybrid solver that replaces the coupled temperature/pressure equation with a trained operator network inside an FEM framework reduces compute by 35-43% while keeping field errors under 5% on unseen loads in thermoel...
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Solved in Unit Domain: JacobiNet for Differentiable Coordinate-Transformed PINNs
A learned coordinate-mapping neural network is inserted in front of a PINN so that PDEs on irregular domains can be solved in a unit reference domain with autograd Jacobians and hard boundary constraints, improving ac...
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Multiprecision computing for multistage fractional physics-informed neural networks
A two-stage, multi-scale fPINN is claimed to reach 10^-7 accuracy, but the reported numbers are inconsistent with the L1 discretization error on the coarse grid.
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