A Neural Operator-Based Approach to Symbolic Discovery of PDEs

Researchers propose NOMTO, a framework combining neural operators with symbolic equation discovery to identify governing equations from complex data involving nonlocal operators and memory effects. This advancement extends traditional symbolic discovery methods beyond local derivatives, enabling discovery of more realistic physical and mathematical models.

The NOMTO framework addresses a fundamental limitation in computational science: discovering governing equations from data when those equations involve nonlocal interactions, coupled field dynamics, or temporal memory. Traditional symbolic discovery tools rely on libraries of local differential operators and algebraic functions, which inadequately represent many real-world physical systems. By embedding pretrained neural operators as learnable nodes within symbolic networks, NOMTO bridges the gap between the interpretability of symbolic approaches and the flexibility of neural methods.

This work builds on the Equation Learner architecture, which uses differentiable computational graphs to find sparse symbolic expressions. The innovation lies in substituting fixed neural operator surrogates for operations that resist simple mathematical expression. The method proves effective on benchmarks involving nonlocal spatial kernels, auxiliary field equations, and integral terms representing memory effects—common features in fluid dynamics, materials science, and climate modeling.

For the scientific computing and machine learning community, NOMTO potentially accelerates physics discovery by automating equation identification in complex systems where manual derivation is infeasible. The ability to recover compact, interpretable equations containing nonlocal operators has implications for scientific reproducibility and theoretical understanding. However, the framework's practical impact depends on the availability of sufficient training data and computational resources to pretrain high-quality neural operators.

Future developments should focus on reducing computational overhead, improving generalization across different operator types, and validating the method on industrial-scale problems. The intersection of neural networks and symbolic computation continues proving fertile ground for advancing scientific discovery capabilities.

• →NOMTO extends symbolic equation discovery to handle nonlocal operators and memory effects beyond traditional local derivative libraries.
• →The framework combines differentiable computational graphs with pretrained neural operator surrogates to achieve interpretable model discovery.
• →Successfully recovers compact governing equations for complex systems involving field couplings and temporal memory effects.
• →Bridges the gap between neural network flexibility and symbolic equation interpretability for scientific computing applications.
• →Advancement has implications for physics discovery in fluid dynamics, materials science, and climate modeling domains.