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

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.

Figures (4)

• Figure 1: Stage 1 noise-sensitivity sweep across the three 1D benchmark cases. Mean$\pm$standard deviation over three seeds for the neural Stage 1 closure errors. In all cases, both diffusion and reaction recovery deteriorate sharply between $1\%$ and $3\%$ observation noise and then flatten toward saturated high-error regimes by $5\%$.
• Figure 2: Neural-to-symbolic error propagation. Stage 2 compression error remains small, while Stage 3 true symbolic error tracks the Stage 1 neural true error. This is the empirical signature of bias preservation.
• Figure 3: Rollout preservation and bias inheritance. Top: symbolic unseen rollout closely follows neural unseen rollout. Bottom: the bias inheritance ratios remain near one, showing that symbolic compression preserves rather than repairs Stage 1 constitutive bias.
• Figure 4: Case Exp observation-degradation breakdown. Left: representative clean, $1\%$, and $5\%$ noise checkpoints for the neural closure errors, together with the symbolic unseen rollout error. Right: temporal stride $2$ alone leaves the mismatch benchmark almost unchanged, while spatial stride $2$ accounts for nearly all of the sparse-setting degradation.

Theorems & Definitions (6)

• Proposition 1: Space-weak consistency
• Proposition 2: Discrete projected identification under sufficient excitation
• Proposition 3: Non-identifiability under limited excitation
• Proposition 4: Approximation bias
• Proposition 5: Bias preservation under symbolic compression
• Remark 1: Finite-time rollout stability