Title: Temporal Normalizing Flows and Continuous Perron-Frobenius operators

Abstract: The generalized Liouville Equation has been rediscovered in the Artificial Intelligence community and has found many applications in probabilistic methods for inference and data generation. In this talk we will discuss in some detail about the Liouville equation and show its intimate connection with uncertainty quantification (UQ). First, we show how the Liouville equation shows up in stochastic sensitivity analysis for dynamical systems. We will derive the sensitivity equations that is crucial in forward UQ and develop probabilistic AI methods to compute sensitivities of statistical quantities to model parameters. This work (with Jehanzeb Chaudhry) has appeared in the proceeding of machine learning research at the International Conference in Artificial Intelligence & Statistics (AISTATS), 2025. If time permits, we will briefly jump into backward UQ and present the Liouville Flow Importance Sampler (LFIS), an innovative generative machine learning model for generating samples from unnormalized density functions. This work (with Yifeng Tian and Yen Ting Lin) has appeared in the proceedings of machine learning research at the International Conference of Machine Learning (ICML), 2024. LFIS adopts a self-learning approach to learn a time-dependent velocity field that deterministically transports samples drawn from a simple initial distribution to a complex target distribution, guided by a prescribed path of annealed distributions. By considering the neural velocity field as an importance sampler, a surprising result emerges: the sample weights can be quantified by accumulating the error along the trajectories driven by neural velocity fields.

Biography:  Nishant Panda is an experienced researcher in uncertainty quantification, statistical machine learning and (in a recent past life) numerical analysis. His doctoral training (Ph.D. from UT, Austin) was on numerical analysis of coupled non-linear PDEs where he developed stable and accurate numerical methods for solving dispersive ocean waves.  At LANL, he developed quantifiable deep learning techniques for scale bridging aimed at learning fracture propagation in materials and is currently working on developing novel statistical learning techniques that require probabilistic models like Normalizing Flows and Optimal transport. As a staff scientist in the information sciences group CAI-3 (Computing and artificial intelligence division), Panda has mentored numerous summer students and co-mentored postdocs. He enjoys classic literature, cooking and is hopelessly addicted to his phone. When he is not frustrated by the current trend of throwing LLMs to everything he likes to think about math pedagogy.