Title: Projective Symbolic Regression: Solving High-Dimensional PDE by Learning from Low-Dimensional Projections
Primary area: neurosymbolic & hybrid AI systems (physics-informed, logic & formal reasoning, etc.)
Keywords: High-dimensional PDEs​, Symbolic Regression, Decomposition Methods, Physics-Informed Machine Learning

Abstract: Symbolic regression (SR) provides a powerful means for uncovering the underlying mathematical structure of physical systems, such as those governed by partial differential equations (PDEs). However, applying SR directly to high-dimensional PDEs remains intractable due to the curse of dimensionality. To address this, we propose Projective Symbolic Regression (PSR), a novel framework that solves high-dimensional PDEs by learning from low-dimensional projections. PSR first generates multiple projections of the PDE solution data by fixing subsets of input variables. Symbolic regression is then applied to each projection to extract compact, localized functional components. These components are subsequently composed into a unified global expression through a higher-level symbolic program. Critically, the final composition is constrained by minimizing the PDE residual error, ensuring physical validity. Empirical results demonstrate that PSR not only improves predictive accuracy over conventional methods but also yields interpretable models that reveal the compositional structure of the underlying physical dynamics.