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

TL;DR

This paper explores neural-symbolic methods for discovering constitutive laws in reaction-diffusion systems, highlighting the impact of bias inheritance during symbolic compression and emphasizing validation over residual minimization.

Contribution

It introduces a three-stage framework combining neural surrogates and symbolic compression, revealing the bias inheritance phenomenon and emphasizing the importance of forward validation.

Findings

Neural surrogates match classical bases under library-matched conditions.

Under function-class mismatch, neural surrogates enable compact symbolic laws.

Bias inheritance causes symbolic errors to closely follow neural surrogate errors.

Abstract

We investigate the data-driven discovery of constitutive closures in nonlinear reaction-diffusion systems with known governing PDE structures. Our objective is to robustly recover diffusion and reaction laws from spatiotemporal observations while avoiding the common pitfall where low residuals or short-horizon predictions are conflated with physical recovery. We propose a three-stage neural-symbolic framework: (1) learning numerical surrogates under physical constraints using a noise-robust weak-form-driven objective; (2) compressing these surrogates into restricted interpretable symbolic families (e.g., polynomial, rational, and saturation forms); and (3) validating the symbolic closures through explicit forward re-simulation on unseen initial conditions. Extensive numerical experiments reveal two distinct regimes. Under matched-library settings, weak polynomial baselines behave as correctly specified reference estimators, showing that neural surrogates do not uniformly outperform classical bases. Conversely, under function-class mismatch, neural surrogates provide necessary flexibility and can be compressed into compact symbolic laws with minimal rollout degradation. However, we identify a critical "bias inheritance" mechanism where symbolic compression does not automatically repair constitutive bias. Across various observation regimes, the true error of the symbolic closure closely tracks that of the neural surrogate, yielding a bias inheritance ratio near one. These findings demonstrate that the primary bottleneck in neural-symbolic modeling lies in the initial numerical inverse problem rather than the subsequent symbolic compression. We underscore that constitutive claims must be rigorously supported by forward validation rather than residual minimization alone.