Solving Physics-Constrained Inverse Problems with Conditional Diffusion Models

Abstract:

Bayesian statistics remain popular for addressing inverse problems, whereby quantities of interest are determined from their noisy and indirect observations.  Bayes’ theorem forms the foundation of this approach by describing the posterior distribution - the conditional density of the quantities of interest given the observation.  The posterior distribution provides a probabilistic characterization of the quantities of interest, which is essential for uncertainty quantification.  Although Markov Chain Monte Carlo (MCMC) methods have long been the workhorse for Bayesian inference, they often encounter significant bottlenecks in large-scale applications.  Metropolis-type MCMC methods do not scale, and gradient-based MCMC methods are computationally expensive.  This talk introduces a novel approach to Bayesian inference through conditional diffusion models, which learn to sample from the posterior distribution by leveraging samples from the joint density of the quantities of interest and observations.  Central to this approach is a neural network-based approximation of the posterior density’s score function, which provides a gradient-informed proposal for scalable MCMC sampling.  Particularly suitable for large-scale applications, conditional diffusion model-based inference overcomes the computational bottlenecks associated with traditional MCMC sampling.  The efficacy of conditional diffusion models is demonstrated through several applications in computational mechanics, where spatially heterogeneous constitutive material properties are inferred from noisy and indirect response measurements.  This talk will also introduce a new derivation for diffusion models from a partial differential equation perspective, discuss the potential pitfalls of diffusion model-based inference, primarily due to data memorization, and suggest methods to regularize against it. 

Speaker’s Bio:
Doctor Agnimitra Dasgupta is a Postdoctoral Research Associate in the Aerospace and Mechanical Engineering Department at the University of Southern California (USC).  He obtained his Ph.D. in Civil Engineering, also from USC.  His research interest lies at the intersection of computational mechanics, uncertainty quantification, and generative machine learning.  His current research focuses on developing novel conditional generative algorithms for data assimilation.  He received the Provost’s Ph.D. Fellowship from USC between 2017 and 2021.  Most recently, he was awarded a travel grant to present his research at the Second United States Association for Computation Mechanics (USACM) Thematic Conference on Uncertainty Quantification for Machine Learning Integrated Physics Modeling (UQ-MLIP 2024).