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PINNacle: A Comprehensive Benchmark of Physics-Informed Neural Networks for Solving PDEs
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While significant progress has been made on Physics-Informed Neural Networks (PINNs), a comprehensive comparison of these methods across a wide range of Partial Differential Equations (PDEs) is still lacking. This study introduces PINNacle, a benchmarking tool designed to fill this gap. PINNacle provides a diverse dataset, comprising over 20 distinct PDEs from various domains, including heat conduction, fluid dynamics, biology, and electromagnetics. These PDEs encapsulate key challenges inherent to real-world problems, such as complex geometry, multi-scale phenomena, nonlinearity, and high dimensionality. PINNacle also offers a user-friendly toolbox, incorporating about 10 state-of-the-art PINN methods for systematic evaluation and comparison. We have conducted extensive experiments with these methods, offering insights into their strengths and weaknesses. In addition to providing a standardized means of assessing performance, PINNacle also offers an in-depth analysis to guide future research, particularly in areas such as domain decomposition methods and loss reweighting for handling multi-scale problems and complex geometry. To the best of our knowledge, it is the largest benchmark with a diverse and comprehensive evaluation that will undoubtedly foster further research in PINNs.
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Cited by 12 Pith papers
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Continual-Learning Physics-Informed Neural Networks for Parameterized Partial Differential Equations
A continual-learning training scheme with Bayesian task selection, dynamic weighting, and sparse physics replay improves accuracy and query efficiency of parameterized physics-informed neural networks on five benchmarks.
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BWLer: Barycentric Weight Layer Elucidates a Precision-Conditioning Tradeoff for PINNs
Adding a barycentric interpolation layer to PINNs lifts their precision ceiling, achieving up to 1e-13 relative error on smooth PDEs, while exposing a tradeoff between accuracy and loss conditioning.
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VideoPDE: Unified Generative PDE Solving via Video Inpainting Diffusion Models
A pixel-space hierarchical video diffusion transformer solves PDE forward, inverse, and sparse-observation tasks by inpainting trajectories, with reported order-of-magnitude error reductions on several 2D benchmark PDEs.
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Physics-informed Temporal Alignment for Auto-regressive PDE Foundation Models
A self-supervised framework that discovers governing equations from short observed data windows and uses them to regularize autoregressive PDE foundation models, improving long-term forecast accuracy.
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PLAID: A Unified Data Model for Machine Learning on Heterogeneous Physics Simulations
PLAID provides a CGNS-based data model, an open-source library, six new simulation datasets, and Hugging Face leaderboards for machine-learning surrogates.
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Physics-Informed Neural Networks for Solving the Two-Dimensional Shallow Water Equations with Terrain Topography and Rainfall Source Terms
PINNs with a reweighted variable-conservation form solve 2D shallow water cases with rainfall and terrain, but the claimed theoretical superiority is only partially supported.
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jinns: a JAX Library for Physics-Informed Neural Networks
The paper introduces jinns, a JAX-native PINN library with claimed first-of-its-kind status and superior speed for inverse problems, supported by a small benchmark study.
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The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning
The Well provides 16 diverse physics simulation datasets totaling 15TB of data, with a unified interface and baselines showing standard surrogate models fail on many of the tasks.
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PINNsAgent: Automated PDE Surrogation with Large Language Models
An LLM-based multi-agent system that automates PINNs hyperparameter optimization, beating random and Bayesian search on 12 of 14 benchmark PDEs but only matching or beating the PINNacle benchmark on 6 of 14.
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Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
A practical recipe to convert common neural architectures into discretization-agnostic neural operators, validated by Navier-Stokes experiments showing cross-resolution generalization of FNO-style models.
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Fast and Flexible Quantum-Inspired Differential Equation Solvers with Data Integration
A QTT finite-difference solver with spline data integration reports logarithmic scaling on Poisson and Burgers equations, with accuracy comparable to classical solvers and PINNs.
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PDE-DKL: PDE-constrained deep kernel learning in high dimensionality
A neural network compresses high-dimensional PDE coordinates into a low-dimensional latent space, where a PDE-constrained Gaussian process achieves accurate solutions and uncertainty estimates on test problems up to 5...
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