Bias Inheritance in Neural-Symbolic Discovery of Constitutive Closures Under Function-Class Mismatch

What mathematical functions do neural network components learn? Symbolic distillation addresses this question by expressing neural network components with interpretable, closed-form mathematical expressions that expose the functional structure learned during training. We develop symbolic distillation as a systematic, architecture-agnostic methodology, and release our approach as the open-source SymTorch package - a PySR-powered library built natively for the PyTorch ecosystem. Applying this methodology across diverse architectures, we find that SymTorch is successful in the automated discovery of physical laws. Specifically, our approach (1) recovers pairwise interaction forces from graph neural networks trained on empirical $n$-body observations, (2) distills the exact closed-form PDE/ODE solutions of multiple physical systems, including the value of constants, from physics-informed neural networks trained on sparse data, and (3) uncovers the chaotic dynamics of the Lorenz system from high-dimensional data, ultimately outperforming the base neural network on downstream prediction tasks. We further demonstrate the utility of our framework for model interpretability by providing an optimized implementation of SLIME - a symbolic extension to the LIME explainability method. SLIME consistently outperforms LIME across predictive metrics across eight popular classification and regression benchmarks, while still providing an interpretable local symbolic model. Lastly, we investigate replacing transformer MLP layers with symbolic surrogates: replacing 1-7 layers with symbolic approximations yields 2-19% throughput improvements and up to 18.7% VRAM reduction, with the resulting hybrid models lying on the Pareto front of throughput versus perplexity among open-source LLMs of comparable scale.

Symbolic discovery of governing equations is a long-standing goal in scientific machine learning, yet a fundamental trade-off persists between interpretability and scalable learning. Classical symbolic regression methods yield explicit analytic expressions but rely on combinatorial search, whereas neural networks scale efficiently with data and dimensionality but produce opaque representations. In this work, we introduce Symbolic Kolmogorov-Arnold Networks (Symbolic-KANs), a neural architecture that bridges this gap by embedding discrete symbolic structure directly within a trainable deep network. Symbolic-KANs represent multivariate functions as compositions of learned univariate primitives applied to learned scalar projections, guided by a library of analytic primitives, hierarchical gating, and symbolic regularization that progressively sharpens continuous mixtures into one-hot selections. After gated training and discretization, each active unit selects a single primitive and projection direction, yielding compact closed-form expressions without post-hoc symbolic fitting. Symbolic-KANs further act as scalable primitive discovery mechanisms, identifying the most relevant analytic components that can subsequently inform candidate libraries for sparse equation-learning methods. We demonstrate that Symbolic-KAN reliably recovers correct primitive terms and governing structures in data-driven regression and inverse dynamical systems. Moreover, the framework extends to forward and inverse physics-informed learning of partial differential equations, producing accurate solutions directly from governing constraints while constructing compact symbolic representations whose selected primitives reflect the true analytical structure of the underlying equations. These results position Symbolic-KAN as a step toward scalable, interpretable, and mechanistically grounded learning of governing laws.

Discovering governing equations from data remains challenging when the underlying dynamics involve nonlocal differential operators, field interactions governed by auxiliary equations, or temporal memory effects. We propose Neural Operator-based symbolic Model approximaTion and discOvery (NOMTO), a framework that extends Equation Learner-type symbolic architectures by incorporating pretrained neural operators as nodes in the symbolic network. NOMTO represents candidate equations as sparse differentiable computational graphs that combine algebraic operations with fixed neural operator surrogates pretrained to approximate nonlinear operators. We evaluate the method on model-discovery problems involving nonlocal spatial operators, couplings mediated by auxiliary field equations, and temporal integral terms representing memory effects. The results show that NOMTO can recover compact governing equations containing nonlocal operator terms, thereby extending symbolic model discovery beyond libraries restricted to local derivatives and point-wise algebraic combinations.