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

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.