SetPINNs: Set-based Physics-informed Neural Networks
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Abstract (arXiv)
Physics-Informed Neural Networks (PINNs) solve partial differential equations using deep learning. However, conventional PINNs perform pointwise predictions that neglect dependencies within a domain, which may result in suboptimal solutions. We introduce SetPINNs, a framework that effectively captures local dependencies. With a finite element-inspired sampling scheme, we partition the domain into sets to model local dependencies while simultaneously enforcing physical laws. We provide a rigorous theoretical analysis showing that SetPINNs yield unbiased, lower-variance estimates of residual energy and its gradients, ensuring improved domain coverage and reduced residual error. Extensive experiments on synthetic and real-world tasks show improved accuracy, efficiency, and robustness.
Key facts. Cited by 2 papers in the map (data through 14 Oct 2026). Builds on 28 of its 57 references that are in the map. First posted on arXiv 30 Sep 2024.
Builds on
28 of 57 references.
References · 28
- 2017Attention Is All You Needcited by 60,361
- 2017Deep Setscited by 1,421
- 2016Optimization Methods for Large-Scale Machine Learningcited by 1,150
- 2017Visualizing the Loss Landscape of Neural Netscited by 1,130
- 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
- 2020When and why PINNs fail to train: A neural tangent kernel perspectivecited by 634
- 2017Solving high-dimensional partial differential equations using deep learningcited by 556
- 2021Physics-informed neural networks (PINNs) for fluid mechanics: A reviewcited by 473
- 2019Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Datacited by 262
- 2022A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networkscited by 236
- 2018Deep Hidden Physics Models: Deep Learning of Nonlinear Partial Differential Equationscited by 235
- 2020Physics-informed learning of governing equations from scarce datacited by 167
16 more
- 2021Finite Basis Physics-Informed Neural Networks (FBPINNs): a scalable domain decomposition approach for solving differential equationscited by 145
- 2022Mitigating Propagation Failures in Physics-informed Neural Networks using Retain-Resample-Release (R3) Samplingcited by 73
- 2023Can Physics-Informed Neural Networks beat the Finite Element Method?cited by 68
- 2023PINNsFormer: A Transformer-Based Framework For Physics-Informed Neural Networkscited by 64
- 2021Universal Approximation of Functions on Setscited by 62
- 2021Learning in Sinusoidal Spaces with Physics-Informed Neural Networkscited by 62
- 2021GaitSet: Cross-view Gait Recognition through Utilizing Gait as a Deep Setcited by 57
- 2021Quadratic Residual Networks: A New Class of Neural Networks for Solving Forward and Inverse Problems in Physics Involving PDEscited by 43
- 2024On the Challenges and Opportunities in Generative AIcited by 31
- 2022Lagrangian PINNs: A causality-conforming solution to failure modes of physics-informed neural networkscited by 26
- 2024RoPINN: Region Optimized Physics-Informed Neural Networkscited by 26
- 2020Machine Learning in Thermodynamics: Prediction of Activity Coefficients by Matrix Completioncited by 19
- 2020Set2Graph: Learning Graphs From Setscited by 16
- 2021How to Avoid Trivial Solutions in Physics-Informed Neural Networkscited by 15
- 2020Semi Conditional Variational Auto-Encoder for Flow Reconstruction and Uncertainty Quantification from Limited Observationscited by 13
- 2023MolSets: Molecular Graph Deep Sets Learning for Mixture Property Modelingcited by 2
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