SetPINNs: Set-based Physics-informed Neural Networks

2024-09-30

arXiv.org

AI Summary

Traditional physics-informed neural networks (PINNs) rely on pointwise predictions, neglecting spatiotemporal local correlations, leading to trivial solutions and poor generalization. To address this, we propose Set-PINN—a set-based physics-informed neural network that pioneers the integration of finite element principles into the PINN framework for region-level PDE modeling. Our method introduces three key components: (i) set-structured input representation, (ii) region-weighted residual loss, and (iii) hard physical constraint embedding, all supported by a finite-element-inspired mesh-based sampling strategy. Notably, Set-PINN is the first approach to systematically incorporate hard constraints within CNN- and Transformer-based architectures. Experiments demonstrate substantial improvements: average error reductions of 12–37% across diverse PDE benchmarks, 2.1× faster convergence, and robust suppression of divergence and trivial-solution failure in real-world fluid dynamics and heat conduction systems.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Hardware-aware ML
• Computer Vision: Low Level & Physics-based Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Physics-Informed Neural Networks (PINNs) have emerged as a promising method for approximating solutions to partial differential equations (PDEs) using deep learning. However, PINNs, based on multilayer perceptrons (MLP), often employ point-wise predictions, overlooking the implicit dependencies within the physical system such as temporal or spatial dependencies. These dependencies can be captured using more complex network architectures, for example CNNs or Transformers. However, these architectures conventionally do not allow for incorporating physical constraints, as advancements in integrating such constraints within these frameworks are still lacking. Relying on point-wise predictions often results in trivial solutions. To address this limitation, we propose SetPINNs, a novel approach inspired by Finite Elements Methods from the field of Numerical Analysis. SetPINNs allow for incorporating the dependencies inherent in the physical system while at the same time allowing for incorporating the physical constraints. They accurately approximate PDE solutions of a region, thereby modeling the inherent dependencies between multiple neighboring points in that region. Our experiments show that SetPINNs demonstrate superior generalization performance and accuracy across diverse physical systems, showing that they mitigate failure modes and converge faster in comparison to existing approaches. Furthermore, we demonstrate the utility of SetPINNs on two real-world physical systems.

Problem

• Physics-Informed Neural Networks
• Complex Mathematical Problems
• Local Data Correlation

Innovation

• SetPINNs
• Physics-Informed Neural Networks
• Finite Element Method