OrthoReg: Orthogonal Regularization for Hybrid Symbolic-Neural Dynamical Systems

Symbolic Regression (SR) seeks to find succinct mathematical expressions that represent the fundamental relationships within data, providing interpretability and scientific understanding that exceeds that of black-box models. Nevertheless, traditional methods like Genetic Programming face challenges with scalability and are highly sensitive to noise, while sparse regression techniques such as SINDy rely significantly on predetermined feature libraries. In this work, we present a Neural Symbolic Regression (NSR) framework that treats neural networks as functional preconditioners for symbolic discovery. Our approach uses a decoupled pipeline: a neural network first learns a smooth, noise-robust approximation of the target function in an interaction- aware nonlinear feature space. LASSO is then applied to extract sparse, interpretable closed-form expressions. To improve predictive accuracy and symbolic fidelity by integrating distributed hyperparameter optimization with Ray Tune and ASHA scheduling. Experiments on the Nguyen benchmark suite show that our approach consistently outperforms SINDy and non-tuned neural baselines in RMSE, noise robustness, and out-of-distribution generalization. Ablation studies confirm the significance of feature interactions, neural depth, and tuning strategies. In general, this study presents a scalable and understandable neural-symbolic framework, creating a solid link between neural approximation and the discovery of sparse equations for scientific machine learning.

Machine learning has become a powerful tool for discovering governing laws of dynamical systems from data. However, most existing approaches degrade severely when observations are sparse, noisy, or irregularly sampled. In this work, we address the problem of learning symbolic representations of nonlinear Hamiltonian dynamical systems under extreme data scarcity by explicitly incorporating physical structure into the learning architecture. We introduce Adaptable Symplectic Recurrent Neural Networks (ASRNNs), a parameter-cognizant, structure-preserving model that combines Hamiltonian learning with symplectic recurrent integration, avoiding time derivative estimation, and enabling stable learning under noise. We demonstrate that ASRNNs can accurately predict long-term dynamics even when each training trajectory consists of only two irregularly spaced time points, possibly corrupted by correlated noise. Leveraging ASRNNs as structure-preserving data generators, we further enable symbolic discovery using independent regression methods (SINDy and PySR), recovering exact symbolic equations for polynomial systems and consistent polynomial approximations for non-polynomial Hamiltonians. Our results show that such architectures can provide a robust pathway to interpretable discovery of Hamiltonian dynamics from sparse and noisy data.

Joint-embedding predictive architectures learn abstract states by predicting target embeddings from context embeddings, but their transition models are typically opaque neural maps. We introduce SJEPA, a reconstruction-free JEPA framework that learns predictive representations whose induced dynamics admit compact symbolic descriptions. Its hybrid transition combines a symbolic law with a regularised neural correction for dynamics outside the selected grammar. The central principle is to learn the simplest adequate dynamics: representation constraints preserve informative, non-collapsed predictive coordinates, while operator compression favours low-complexity symbolic-neural transitions that remain predictively adequate. We formalise this principle through induced-dynamics complexity, analyse predictive-coordinate non-identifiability, and show that unconstrained operator compression creates a direct shortcut to representation collapse. The framework supports both alternating representation-equation learning and symbolic dynamics fitted to fixed representations. In controlled pendulum experiments, joint learning discovers substantially simpler symbolic dynamics with lower long-horizon rollout error and divergence than post-hoc fitting, while an unconstrained one-step diagnostic realises the predicted collapse shortcut. Under grammar misspecification, correction regularisation preserves the representable symbolic mechanism and directs the neural component towards residual dynamics. The results expose a controllable trade-off among predictive fidelity, representation quality, symbolic parsimony, and symbolic-neural allocation.