Integrating prior knowledge in equation discovery: Interpretable symmetry-informed neural networks and symbolic regression via characteristic curves

TL;DR

This work addresses the identifiability challenges in data-driven equation discovery by adopting a modular CC-based framework that partitions dynamics into univariate constitutive relations learned as CCs. It extends NN-CC with symmetry constraints and symbolic-regression post-processing, and demonstrates improved recovery of underlying laws in chaotic Duffing dynamics and non-smooth stick-slip friction, across NN-CC, SINDy-CC, and SR-CC variants. The study shows that incorporating prior structure and parsimony reduces overfitting to noise, enhances forward-prediction reliability within training domains, and yields interpretable constitutive forms suitable for physical insight. Overall, the CC-based approach offers a solver-agnostic, physics-informed path to robust, interpretable system identification, with promising avenues toward higher dimensions, automated symmetry discovery, and probabilistic uncertainty quantification.

Abstract

Data-driven equation discovery aims to reconstruct governing equations directly from empirical observations. A fundamental challenge in this domain is the ill-posed nature of the inverse problem, where multiple distinct mathematical models may yield similar errors, thus complicating model selection and failing to guarantee a unique representation of the underlying mechanisms. This issue can be addressed by incorporating inductive biases to constrain the search space and discard the undesirable models. The characteristic curves-based (CCs) framework offers a modular approach ideally suited to this aim. This approach is based on the specification of structural families that possess provable identifiability properties. Crucially, this framework enables practitioners to embed domain expertise directly into the learning process and facilitates the integration of diverse post-processing tools. In this work, we build upon the recent neural network implementation of this formalism (NN-CC), which benefits from the universal approximation capabilities of NNs. Specifically, we extend NN-CC by introducing two inductive biases: (i) symmetry constraints and (ii) post-processing with symbolic regression. Using a chaotic Duffing oscillator and a discontinuous stick-slip model under varying Gaussian noise levels, we show how these extensions systematically improve the discovery process. We also analyze the integration of sparse and symbolic regression (using SINDy and PySR) into the CC-based formalism. These extensions (SINDy-CC and SR-CC) consistently show improvements as prior information is incorporated. By enabling the integration of prior or hypothesized knowledge into the learning and post-processing stages, the CC-based formalism emerges as a promising candidate to address identifiability issues in purely data-driven methods, advancing the goal of interpretable and reliable system identification.

Figures (17)

• Figure 1: Schematic workflow of the proposed CC-based formalism. Starting from raw data (a), the practitioner selects the variables that will intervene in the model (a), they define a equation structure (c) where unknown physical mechanisms CCs are approximated by learnable functions (e.g., Neural Networks, polynomials, or sparse regression terms). The optimization process (f) minimizes a loss function (d) that balances data reconstruction with physical priors and symmetries (e). The result (g) is a set of interpretable functions (e.g., damping $f_1$ and stiffness $f_2$) that accurately predict system dynamics via numerical integration (i) and (j). An optional stage of post-processing of the obtained CCs can be also performed (h). The workflow is exemplified with NNs, but other basis functions for the CCs are also analyzed.
• Figure 2: Schematic diagram of the optimization process for the CC-based formalism applied to a family of second order systems. This schematic is parameterized using NNs (NN-CC method), but other basis functions for the CCs such as polynomials and sparse regression are also analyzed.
• Figure 3: System identification workflow based on the Duffing example. (a) and (b) show the input data: dynamical variable $x(t)$ and driving force $F_{ext}(t)$ with SNR = 20 dB, respectively. (c) and (d) show the identified models obtained using the NN-CC$_{\text{+sym+post-SR}}$ method: $f_1(\dot{x})$ and $f_2(x)$, respectively. The gray zones indicate the range of training dataset values.
• Figure 4: Deviation of predicted CCs from theoretical values, NN$_2(x)-f_2^{\text{th}}(x)$. Panels (a–c) correspond to NN-CC and panels (d–f) to NN-CC$_\text{+sym}$ and NN-CC$_\text{+sym+post-SR}$: (a,d) single training run; (b,e) 100 independent training instances; (c,f) 10 noise realizations with one training run per realization. In the 100-run cases, solid lines indicate the mean, while shaded regions show the [min, max] interval across runs.
• Figure 5: RMSE analysis for the identified CCs ($f_1$ and $f_2$) for the Duffing system. Panels (a) and (b): NN-CC variants; (c) and (d): Polynomial basis variants; (e) and (f): SR variants.