Neuro-Symbolic AI for Analytical Solutions of Differential Equations

Analytical solutions of differential equations offer exact insights into fundamental behaviors of physical processes. Their application, however, is limited as finding these solutions is difficult. To overcome this limitation, we combine two key insights. First, constructing an analytical solution requires a composition of foundational solution components. Second, iterative solvers define parameterized function spaces with constraint-based updates. Our approach merges compositional differential equation solution techniques with iterative refinement by using formal grammars, building a rich space of candidate solutions that are embedded into a low-dimensional (continuous) latent manifold for probabilistic exploration. This integration unifies numerical and symbolic differential equation solvers via a neuro-symbolic AI framework to find analytical solutions of a wide variety of differential equations. By systematically constructing candidate expressions and applying constraint-based refinement, we overcome longstanding barriers to extract such closed-form solutions. We illustrate advantages over commercial solvers, symbolic methods, and approximate neural networks on a diverse set of problems, demonstrating both generality and accuracy.

RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains

Learning the solution operators of PDEs on arbitrary domains is challenging due to the diversity of possible domain shapes, in addition to the often intricate underlying physics. We propose an end-to-end graph neural network (GNN) based neural operator to learn PDE solution operators from data on point clouds in arbitrary domains. Our multi-scale model maps data between input/output point clouds by passing it through a downsampled regional mesh. Many novel elements are also incorporated to ensure resolution invariance and temporal continuity. Our model, termed RIGNO, is tested on a challenging suite of benchmarks, composed of various time-dependent and steady PDEs defined on a diverse set of domains. We demonstrate that RIGNO is significantly more accurate than neural operator baselines and robustly generalizes to unseen spatial resolutions and time instances.

Vandermonde Neural Operators

Fourier Neural Operators (FNOs) have emerged as very popular machine learning architectures for learning operators, particularly those arising in PDEs. However, as FNOs rely on the fast Fourier transform for computational efficiency, the architecture can be limited to input data on equispaced Cartesian grids. Here, we generalize FNOs to handle input data on non-equispaced point distributions. Our proposed model, termed as Vandermonde Neural Operator (VNO), utilizes Vandermonde-structured matrices to efficiently compute forward and inverse Fourier transforms, even on arbitrarily distributed points. We present numerical experiments to demonstrate that VNOs can be significantly faster than FNOs, while retaining comparable accuracy, and improve upon accuracy of comparable non-equispaced methods such as the Geo-FNO.