Data-driven symbolic discovery of nonlinear dynamics

Abstract

Nonlinear dynamical systems are omnipresent in nature, commonly seen in many disciplines such as physics, biology, chemistry, climate science, and engineering. Discovering the analytical expressions of underlying physics that govern the systems from their measurement data is essential for understanding the behaviors of the observed nonlinear dynamics, which might be complex, or even chaotic. Directly distilling the nonlinear governing equations from limited and noisy measurement data has long been vital but challenging. To tackle this fundamental issue, this dissertation introduces a novel Physics-informed Spline Learning (PiSL) framework to discover parsimonious governing equations for nonlinear dynamics, based on sparsely sampled noisy data. Specifically, splines are employed to locally interpolate the dynamics and perform analytical differentiation, feeding the discovery of underlying equations in form of either a linear interpolation of candidate terms or a symbolic-activated neural network model. The physics residual in turn informs the spline learning. The synergy between splines and discovered governing equations produces great robustness against high-level data sparsity and noise. Subsequently, a hybrid sparsity-promoting alternating direction optimization strategy is developed for fine-tuning the coefficients with a sparsity enforcement approach to obtain a parsimonious structure of discovered governing equations. The effectiveness and supremacy of the proposed PiSL architectures have been demonstrated by several numerical and experimental examples, in comparison with two state-of-the-art methods serving as baselines. Meanwhile, this dissertation re-envisions the data-driven nonlinear dynamics discovery tasks by casting them into symbolic regression problems. Under this scheme, a symbolic regressor is established to identify the differential equations that best describe the underlying governing physical laws without fixed forms of expressions as a starting point, leveraging great flexibility (i.e, free combination of mathematical operations and symbols) in model selection. This dissertation develops an innovative Symbolic Reinforcement Learning (SRL) machine to discover the mathematical structure of equations based on sparse and noisy measurement data. The central concept is to (i) interpret mathematical operations and variables by symbols following certain grammar rules, (ii) establish the symbolic reasoning of equations via expression trees, and (iii) develop an environment for the reinforcement learning (RL) agent to explore optimal expression trees based on data. In particular, the RL agent is able to obtain an optimistic computational policy through the traversal of expression trees, featuring the one that maps to the optimal arithmetic expression of the underlying equation. The robustness of the SRL machine is demonstrated by examples of symbolic regression and the identification of nonlinear dynamics (both numerically and experimentally). Salient features of the proposed framework include search flexibility and enforcement of parsimony for discovered equations. It offers a new perspective to finding interpretable and generalizable symbolic models for facilitating cross-disciplinary data-driven scientific discovery. Moreover, this dissertation presents an amelioration to the proposed RL-based symbolic regressor, the Symbolic Physics Learner (SPL) machine. The SPL machine employs Monte Carlo tree search (MCTS) algorithm, which is featured by a sound mathematical underpinning for the trade-off between exploration and exploitation. A few adjustments to the conventional MCTS algorithm are made to better fit the symbolic regression and nonlinear dynamics discovery problems. Consequently, the SPL machine is capable of efficiently uncovering the best path to formulate the complex mathematical expressions and governing equations of the target dynamical system. The efficacy and superiority of the PSL machine are demonstrated by numerical examples including the classic Nguyen symbolic regression benchmarks, the tasks of physics law discovery from experiment-measured data, and the nonlinear dynamics discovery experiments, in comparison with state-of-the-art methods.--Author's abstract