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Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Sampling
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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 12 Pith papers
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Curriculum Learning of Physics-Informed Neural Networks based on Spatial Correlation
A spatially correlated curriculum learning framework for PINNs using causal weights, low-frequency bridges, and adaptive reweighting to reduce training failures on spatially coupled BVPs.
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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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When PINNs Go Wrong: Pseudo-Time Stepping Against Spurious Solutions
PINNs fail on spurious solutions admitted by the residual loss; adaptive pseudo-time stepping with Jacobian-based step selection improves accuracy and robustness on PDE benchmarks.
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Deep Learning Alternatives of the Kolmogorov Superposition Theorem
ActNet is a new KST-based neural network that outperforms KANs and competes with MLPs in PINN benchmarks for PDE simulation tasks.
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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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Differentiable Chemistry in PINNs for Solving Parameterized and Stiff Reaction Systems
A differentiable chemistry solver is added to PINNs along with parameterized network architecture and stiffness-tailored residual weighting to solve initial/boundary value problems, inverse parameter identification, a...
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Physics-informed neural networks for form-finding of unilateral membrane structures
PINNs with hard and soft boundary enforcement solve membrane form-finding PDEs to accuracy comparable with FEM, with hard-BC yielding smaller boundary errors.
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On the training of physics-informed neural operators for solving parametric partial differential equations
Empirical benchmarks indicate CViT-based PINOs with adapted PINN mitigations can match or exceed data-driven neural operators on parametric PDE tasks.
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Integral regularization PINNs for evolution equations
IR-PINNs improve long-time accuracy for evolution equations by enforcing integral constraints over time sub-intervals and using adaptive collocation point sampling.
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A numerical study into neural network surrogate model performance for uncertainty propagation
Numerical study comparing feedforward NN and DeepONet with data-driven and physics-informed losses on stochastic heat equation, highlighting larger errors at distribution tails due to extrapolation.
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