Theory and Computation of PDE Team

Siyao Yang

Monday, September 21, 2026 3:00 PM - 4:00 PM

A3-4-312

Zoom 518 868 7656 (BIMSA)

High-dimensional probability distributions arise throughout scientific computing, yet their estimation, representation, and evaluation are often hindered by the curse of dimensionality. In this talk, I will present a tensor-train-based framework for learning high-dimensional densities from samples. The method first smooths the empirical distribution to reduce its large variance, compresses the resulting coefficient tensor using efficient tensor-train algorithms, and then applies a deconvolution step to recover a functional density estimator. The proposed method can achieve computational complexity linear in both the dimension and the number of samples. Its performance is demonstrated on several tasks, including high-dimensional Boltzmann distribution estimation and image generation.

I will then discuss an application of this framework to molecular dynamics called TT-Metadynamics. In this method, the growing sum of Gaussian bias potentials generated during metadynamics is periodically compressed into a functional tensor train. This avoids exponential grid storage and prevents the cost of evaluating the bias potential from increasing with simulation time. The method enables free-energy exploration with up to 14 collective variables and performs favorably compared with standard metadynamics in high-dimensional examples.