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

TL;DR

NMIPS introduces a multitask symbolic regression framework for PDE families, combining multifactorial optimization with an affine knowledge-transfer mechanism to uncover analytical solutions that span parameterized PDEs. By encoding solutions as expression trees (via C-ADF/Karva) in a unified chromosome and guiding search with data-fit and PDE-consistency losses, NMIPS discovers interpretable closed-form expressions across all tasks in a family. The affine transfer module aligns latent representations across related PDEs, boosting search efficiency and solution quality. Empirical results across six representative PDEs show NMIPS achieves higher accuracy and significantly lower computational costs than baselines, with strong robustness to noise, highlighting its practical potential for rapid, interpretable discovery of PDE solutions.

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.

Figures (7)

• Figure 1: Numerical solutions’ landscapes for PDEs with different parameters reveal that equations belonging to the same family demonstrate analogous analytical solutions.
• Figure 2: Schematic comparison of different paradigms for solving PDEs. (a) Physics-informed Neural Networks embed physical laws by constraining the network training with PDE residuals. Despite integrating physics, they yield uninterpretable numerical approximations and require retraining for each instance, operating as instance-specific black-box models. (b) Neural Operator achieves generalizability by learning mappings between function spaces; however, they essentially remain black-box models that output numerical solutions without analytical forms. (c) Genetic Programming Symbolic Regression offers interpretability by searching for analytical expressions, yet it is constrained to solving one specific problem at a time. (d) Our method constructs a unified search space and leverages a transfer mechanism between tasks, realizing a paradigm that is both multitasking capable and interpretable.
• Figure 3: The overall framework of the proposed NMIPS. (a) Multifactorial PDE solving framework: This part constructs a unified encoding space to execute operations such as crossover, mutation, and selection. It embeds a knowledge transfer module to enable the efficient solving of multiple tasks within a PDE family. (b) Affine Transformation-based Knowledge Transfer: This mechanism extracts statistics of each population and uses multi-layer perceptrons to learn scaling and shifting parameters. It performs an affine transformation across populations to achieve efficient knowledge sharing between tasks.
• Figure 4: Structure of the C-ADF.
• Figure 5: Example of a chromosome includes one main function and one ADF.