Inferring physical laws from data is a central challenge in science and engineering, including but not limited to healthcare, physical sciences, biosciences, social sciences, sustainability, climate, and robotics. Deep networks offer high-accuracy results but lack interpretability, prompting interest in models built from simple components. The Sparse Identification of Nonlinear Dynamics (SINDy) method has become the go-to approach for building such modular and interpretable models. SINDy leverages sparse regression with L1 regularization to identify key terms from a library of candidate functions. However, SINDy's choice of candidate library and optimization method requires significant technical expertise, limiting its widespread applicability. This work introduces Al-Khwarizmi, a novel agentic framework for physical law discovery from data, which integrates foundational models with SINDy. Leveraging LLMs, VLMs, and Retrieval-Augmented Generation (RAG), our approach automates physical law discovery, incorporating prior knowledge and iteratively refining candidate solutions via reflection. Al-Khwarizmi operates in two steps: it summarizes system observations-comprising textual descriptions, raw data, and plots-followed by a secondary step that generates candidate feature libraries and optimizer configurations to identify hidden physics laws correctly. Evaluating our algorithm on over 198 models, we demonstrate state-of-the-art performance compared to alternatives, reaching a 20 percent increase against the best-performing alternative.

Nonlinear dynamics is a pervasive phenomenon observed in scientific and engineering disciplines. However, the task of deriving analytical expressions to describe nonlinear dynamics from limited data remains challenging. In this paper, we shall present a novel deep symbolic learning method called the "finite expression method" (FEX) to discover governing equations within a function space containing a finite set of analytic expressions, based on observed dynamic data. The key concept is to employ FEX to generate analytical expressions of the governing equations by learning the derivatives of partial differential equation (PDE) solutions through convolutions. Our numerical results demonstrate that our FEX surpasses other existing methods (such as PDE-Net, SINDy, GP, and SPL) in terms of numerical performance across a range of problems, including time-dependent PDE problems and nonlinear dynamical systems with time-varying coefficients. Moreover, the results highlight FEX's flexibility and expressive power in accurately approximating symbolic governing equations.