Neuro-Symbolic Multitasking: A Unified Framework for Discovering Generalizable Solutions to PDE Families

2026-02-12

AI Summary

Traditional numerical methods for solving families of partial differential equations (PDEs) require independent computations for each instance, incurring high computational costs; while existing machine learning approaches offer efficiency, they typically lack interpretable analytic expressions. This work proposes NMIPS, a neural-symbolic multitask framework that, for the first time, integrates neural-symbolic computation with multitask optimization to uniformly discover interpretable analytic solutions across a PDE family via an affine transfer mechanism, thereby eliminating redundant solves. By synergistically combining symbolic regression with multi-factor optimization, the method achieves up to a 35.7% improvement in accuracy over baseline approaches across multiple test cases while yielding scientifically interpretable closed-form expressions.

Technology Category

• Machine Learning: Neuro-Symbolic Learning
• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Multiagent Systems: Distributed Problem Solving

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

Abstract

Solving Partial Differential Equations (PDEs) is fundamental to numerous scientific and engineering disciplines. A common challenge arises from solving the PDE families, which are characterized by sharing an identical mathematical structure but varying in specific parameters. Traditional numerical methods, such as the finite element method, need to independently solve each instance within a PDE family, which incurs massive computational cost. On the other hand, while recent advancements in machine learning PDE solvers offer impressive computational speed and accuracy, their inherent ``black-box"nature presents a considerable limitation. These methods primarily yield numerical approximations, thereby lacking the crucial interpretability provided by analytical expressions, which are essential for deeper scientific insight. To address these limitations, we propose a neuro-assisted multitasking symbolic PDE solver framework for PDE family solving, dubbed NMIPS. In particular, we employ multifactorial optimization to simultaneously discover the analytical solutions of PDEs. To enhance computational efficiency, we devise an affine transfer method by transferring learned mathematical structures among PDEs in a family, avoiding solving each PDE from scratch. Experimental results across multiple cases demonstrate promising improvements over existing baselines, achieving up to a $\sim$35.7% increase in accuracy while providing interpretable analytical solutions.

Problem

• Partial Differential Equations
• PDE families
• analytical solutions
• interpretability
• computational efficiency

Innovation

• Neuro-Symbolic
• Multitasking
• PDE Families
• Affine Transfer
• Interpretable Solutions