Topological Neural Operators

Modeling complex physical systems accurately and efficiently remains a fundamental challenge across science and engineering. Classical numerical methods like finite element and spectral solvers provide high fidelity but are computationally intensive, especially for large-scale or real-time applications. The advent of neural operators (NOs) has revolutionized this landscape by enabling data-driven approximation of PDE solution operators, offering significant speedups while maintaining accuracy. Notable examples include Fourier Neural Operators (FNO) and graph-based neural operators, which have demonstrated impressive results on structured and semi-structured meshes.

However, these existing methods predominantly operate on point clouds or graphs, neglecting the intrinsic geometric and topological nature of physical fields. Physical quantities such as potentials, fluxes, and densities are inherently tied to geometric objects of different dimensions—vertices, edges, faces, and volumes—and their interactions are governed by differential operators like gradient, curl, and divergence. These operators encode conservation laws and physical identities (e.g., div curl = 0), which are crucial for physically consistent modeling. Traditional neural operators often obscure this structure, limiting their ability to enforce physical laws and model multi-physics systems accurately.

To address this, the authors introduce Topological Neural Operators (TNO), a novel framework grounded in discrete exterior calculus (DEC). TNO operates directly on cell complexes, where data is supported on cells of varying ranks, and models interactions through fixed topological operators. This explicit encoding of the geometric and topological structure ensures the model respects physical conservation laws and compatibility conditions. The architecture separates where information flows—dictated by the fixed operators—from how it is transformed—learned through neural networks. This design guarantees that the learned features align with the physical support of the quantities involved.

Furthermore, the paper extends this idea with Hierarchical TNO (HTNO), which incorporates learned coarse complexes to facilitate long-range and topology-dependent information propagation. This multi-scale approach enables the model to handle large, complex geometries effectively, capturing both local details and global structures. The authors validate their framework on a suite of PDE benchmarks, including irregular geometries and multi-physics problems, demonstrating significant improvements over existing methods. For instance, in complex flow simulations, error reductions of over 20% were observed, with better physical consistency and generalization across discretizations.

This work represents a substantial advancement in physics-informed machine learning, bridging the gap between topological geometry and neural PDE solvers. By embedding the physical laws directly into the architecture through DEC, TNO and HTNO set the stage for scalable, accurate, and physically consistent simulations in diverse scientific and engineering domains. The approach opens new avenues for tackling previously intractable problems involving complex geometries, multi-physics coupling, and multi-scale phenomena, promising a transformative impact on computational science.