Symbolic Recovery of Differential Equations: The Identifiability Problem
Symbolic recovery of differential equations is the ambitious attempt at automating the derivation of governing equations with the use of machine learning techniques. In contrast to classical methods which assume the structure of the equation to be known and focus on the estimation of specific parameters, these algorithms aim to learn the structure and the parameters simultaneously. While the uniqueness and, therefore, the identifiability of parameters of governing equations are a well-addressed problem in the field of parameter estimation, it has not been investigated for symbolic recovery. However, this problem should be even more present in this field since the algorithms aim to cover larger spaces of governing equations. In this paper, we investigate under which conditions a solution of a differential equation does not uniquely determine the equation itself. For various classes of differential equations, we provide both necessary and sufficient conditions for a function to uniquely determine the corresponding differential equation. We then use our results to devise numerical algorithms aiming to determine whether a function solves a differential equation uniquely. Finally, we provide extensive numerical experiments showing that our algorithms can indeed guarantee the uniqueness of the learned governing differential equation, without assuming any knowledge about the analytic form of function, thereby ensuring the reliability of the learned equation.
Code (1)
Tasks
parameter estimationSimilar Papers 제목 키워드 기반
Robust identifiability for symbolic recovery of differential equations
Recent advancements in machine learning have transformed the discovery of physical laws, moving from manual derivation to data-driven methods that simultaneously learn both the structure and parameters of governing equat…
parameter estimationSymbolic Discovery of Stochastic Differential Equations with Genetic Programming
Automated scientific discovery aims to improve scientific understanding through machine learning. A central approach in this field is symbolic regression, which uses genetic programming or sparse regression to learn inte…
Towards Identifiability of Interventional Stochastic Differential Equations
We study identifiability of stochastic differential equation (SDE) models under multiple interventions. Our results give the first provable bounds for unique recovery of SDE parameters given samples from their stationary…
LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search
Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over …
Bayesian polynomial neural networks and polynomial neural ordinary differential equations
Symbolic regression with polynomial neural networks and polynomial neural ordinary differential equations (ODEs) are two recent and powerful approaches for equation recovery of many science and engineering problems. Howe…
Bayesian InferencePolynomial Neural NetworksSymbolic RegressionVariational Inference