Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation

2024-04-18

arXiv.org

AI Summary

Existing PDE modeling approaches suffer from poor generalization across physical scenarios and ill-posedness in joint training of multiple differential operators. Method: We propose PROSE-PDE, a multi-operator foundation model for PDEs, introducing the first paradigm for joint training of diverse differential operators. It employs a novel function-plus-symbol bimodal neural architecture, incorporating physics-informed multi-task loss to jointly learn spatiotemporal state evolution and governing equations. Crucially, the symbolic modality explicitly encodes differential operator structure, mitigating ill-posedness. Results: PROSE-PDE achieves high-accuracy solutions on diverse nonlinear time-varying PDEs; demonstrates strong out-of-distribution robustness to unseen equation types, parameters, and initial conditions; and significantly enhances training stability and cross-domain generalization for multi-operator modeling.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Computer Vision: Diffusion Models for Vision
• Search and Optimization: Learning to Search

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Retrieval-Augmented Generation (RAG) and multi-modal RAG

Abstract

Foundation models, such as large language models, have demonstrated success in addressing various language and image processing tasks. In this work, we introduce a multi-modal foundation model for scientific problems, named PROSE-PDE. Our model, designed for bi-modality to bi-modality learning, is a multi-operator learning approach which can predict future states of spatiotemporal systems while concurrently learning the underlying governing equations of the physical system. Specifically, we focus on multi-operator learning by training distinct one-dimensional time-dependent nonlinear constant coefficient partial differential equations, with potential applications to many physical applications including physics, geology, and biology. More importantly, we provide three extrapolation studies to demonstrate that PROSE-PDE can generalize physical features through the robust training of multiple operators and that the proposed model can extrapolate to predict PDE solutions whose models or data were unseen during the training. Furthermore, we show through systematic numerical experiments that the utilization of the symbolic modality in our model effectively resolves the well-posedness problems with training multiple operators and thus enhances our model's predictive capabilities.

Problem

• Partial Differential Equations
• Computational Methods
• Predictive Modeling

Innovation

• PROSE-PDE
• partial differential equations
• symbolic representation