Houman Owhadi, Ph.D.

Professor
Applied and Computational Mathematics and Control and Dynamical Systems
California Institute of Technology (Caltech)

 

Abstract: Many challenges in science and engineering involve discovering functional relationships between variables and uncovering the underlying graphical structures that govern complex systems. These challenges can be categorized into three levels of increasing complexity:

• Type 1: Approximate an unknown function (e.g., a stress-strain relationship) using input/output data.
• Type 2: Represent a system as a graph of variables and functions (some unknown) indexed by nodes and edges. Given partial observations (e.g., boundary conditions or sensor measurements), estimate unobserved variables and unknown functions while ensuring consistency with the physics and dependencies encoded in the graph’s structure.
• Type 3: When the graph structure itself is unknown, use partial observations to infer the structure (e.g., interdependencies in a mechanical system) and approximate the unknown functions.

Examples of Type 2 problems include solving nonlinear partial differential equations (PDEs) and learning (possibly stochastic and/or differential) governing equations from limited data. Type 3 problems encompass discovering dependencies in mechanical systems, identifying chemical reaction networks, determining relationships in protein-signaling networks, and, more generally, data-driven model discovery.

Although Gaussian Process (GP) methods are sometimes perceived as a well-founded but old technology limited to Type 1 curve fitting, they can be generalized into an interpretable framework for solving Type 2 and Type 3 problems all while maintaining the simple and transparent theoretical and computational guarantees of kernel methods. This extension leverages the variance decomposition and nonlinear sensitivity analysis properties of GPs, which lack natural counterparts in neural network methods. These properties allow GPs to seamlessly integrate traditional physics-based modeling with data-driven techniques, offering engineers a transparent, computationally efficient, and theoretically grounded approach to model discovery, uncertainty quantification, and predictive analysis.

Biosketch: Houman Owhadi is a professor of applied and computational mathematics and control and dynamical systems at the California Institute of Technology. His expertise includes uncertainty quantification, numerical approximation, statistical inference/learning, data assimilation, stochastic and multiscale analysis, and scientific machine learning. He was a plenary speaker at SIAM CSE 2015 and SIAM UQ 2024 and a tutorial speaker at SIAM UQ 2016. He received the 2019 Germund Dahlquist SIAM Prize. He is a SIAM Fellow (class of 2022) and a Vannevar Busch Fellow (class of 2024).