A Neural Operator-Based Approach to Symbolic Discovery of PDEs

Technical Summary: A Neural Operator-Based Approach to Symbolic Discovery of PDEs

Problem Statement
Existing methods for symbolic model discovery, such as Equation Learner (EQL) and sparse regression approaches (e.g., PDE-FIND), are fundamentally limited by their reliance on libraries of local analytical functions. These methods construct candidate equations using point-wise quantities, including field values, local derivatives, and algebraic combinations. While effective for classical Partial Differential Equations (PDEs), they fail to discover governing laws involving nonlocal interactions, integral operators, auxiliary field equations, or temporal memory effects. Many physical systems—such as those governed by fractional diffusion, coupled field systems, or hereditary viscoelasticity—depend on operators that act on entire function spaces rather than local points, rendering them inaccessible to current symbolic discovery frameworks.

Methodology: NOMTO
The authors propose Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO), a framework that extends the EQL architecture to incorporate pretrained neural operators as symbolic primitives.

• Architecture: NOMTO represents candidate equations as sparse, differentiable computational graphs. Unlike standard EQL, where nodes apply only predefined algebraic or local differential operations, NOMTO allows selected nodes to be implemented as pretrained neural operator surrogates. These surrogates act on discretized functions and return discretized operator outputs on the same grid.
• Separation of Learning and Discovery: A core design principle is the decoupling of operator learning from equation discovery.
  1. Pretraining: Neural operators (specifically Fourier Neural Operators [FNO], Convolutional Neural Operators [CNO], and Fully Connected Neural Networks [FNN]) are trained independently to approximate specific physical operators (e.g., fractional Laplacian, Poisson-gradient map, hereditary memory integrals). These surrogates are then frozen.
  2. Symbolic Optimization: During the discovery phase, the surrogate nodes remain fixed. The optimization process adjusts only the trainable scalar weights connecting the nodes in the computational graph.
• Optimization Strategy: The framework employs a coarse-to-sparse paradigm involving three phases:
  1. Dense Training: The full graph is trained to fit the target dynamics without sparsity constraints.
  2. Sparsification: An $\ell_1$ penalty is added to the loss function, and weights below a certain magnitude are pruned to identify the sparse structure.
  3. Refitting: The pruned graph is retrained without the sparsity penalty to refine the coefficients of the active paths.
• Expression Extraction: After training, negligible weights are removed, and the remaining active graph is translated into an explicit symbolic equation. The resulting expression combines standard algebraic terms with the notation for the pretrained neural operator surrogates (e.g., $\hat{G}^a_m$).

Key Results
NOMTO was evaluated on three distinct physical systems requiring nonlocal or history-dependent operators:

1. Fractional Diffusion: The system involves a fractional Laplacian operator, a nonlocal spatial operator. NOMTO successfully recovered the governing equation $u_t \approx -\hat{\kappa} \hat{G}^a_{L_s}[u]$. The FNO-based surrogate achieved the highest accuracy (Relative MSE $9.47 \times 10^{-6}$) and correctly identified the coefficient $\hat{\kappa} = 1.00$.
2. Euler-Poisson System: This system couples local transport equations with a nonlocal force term derived from an auxiliary Poisson equation. NOMTO recovered both the local convective terms and the nonlocal Poisson-gradient coupling. The CNO surrogate achieved the lowest momentum equation error (Relative MSE $3.245 \times 10^{-4}$), while all architectures correctly identified the symbolic structure and coefficients.
3. Hereditary Viscoelasticity: This case involves a constitutive law with a temporal memory integral. NOMTO successfully discovered the structure $\sigma(t) = E\varepsilon(t) + A\hat{G}^a_H[\varepsilon](t)$, recovering the instantaneous elastic modulus and the memory amplitude with high accuracy across all surrogate types.

In all cases, NOMTO outperformed or matched standard neural network surrogates (FNN) in terms of structural correctness and coefficient recovery, demonstrating that pretrained operators can serve as faithful symbolic proxies for complex physical mechanisms.

Significance and Claims
The paper claims that NOMTO represents a shift from "symbolic discovery of functions" to "symbolic discovery of operator-valued physical laws." By embedding learned operators as fixed, differentiable nodes within a sparse symbolic graph, the framework:

• Expands the Discoverable Class: It extends symbolic discovery beyond local differential libraries to include governing laws with nonlocal spatial operators, auxiliary-field mappings, and temporal memory effects.
• Preserves Interpretability: Despite using neural networks, the final output remains an interpretable symbolic equation where the role of nonlocal mechanisms is explicit.
• Enables Modularity: The separation of operator learning and equation discovery allows operator libraries to be reused across different discovery tasks and facilitates the incorporation of prior physical knowledge through the selection of candidate operators.

The authors acknowledge limitations, noting that the fidelity of the recovered model depends on the accuracy of the surrogate operators and that the construction of the operator library still requires some degree of prior knowledge regarding plausible physical mechanisms. However, the work establishes a pathway for integrating advances in operator learning with interpretable scientific machine learning.