Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation

Modern physics simulation often involves multiple functions of interests, and traditional numerical approaches are known to be complex and computationally costly. While machine learning-based surrogate models can offer significant cost reductions, most focus on a single task, such as forward prediction, and typically lack uncertainty quantification --- an essential component in many applications. To overcome these limitations, we propose Arbitrarily-Conditioned Multi-Functional Diffusion (ACM-FD), a versatile probabilistic surrogate model for multi-physics emulation. ACM-FD can perform a wide range of tasks within a single framework, including forward prediction, various inverse problems, and simulating data for entire systems or subsets of quantities conditioned on others. Specifically, we extend the standard Denoising Diffusion Probabilistic Model (DDPM) for multi-functional generation by modeling noise as Gaussian processes (GP). We propose a random-mask based, zero-regularized denoising loss to achieve flexible and robust conditional generation. We induce a Kronecker product structure in the GP covariance matrix, substantially reducing the computational cost and enabling efficient training and sampling. We demonstrate the effectiveness of ACM-FD across several fundamental multi-physics systems.

Simulating complex physical systems often involves many interacting components, such as pressure, velocity, and temperature. In real-world applications, scientists and engineers may want to predict some components based on others, complete missing information, estimate uncertainties of predictions, or simulate the entire system. Most machine learning models today are designed to do only one of these tasks at a time. This paper introduces ACM-FD (Arbitrarily-Conditioned Multi-Functional Diffusion for Multi-Physics Emulation), a probabilistic model that solves all these problems in one model within a single training. It uses diffusion models to generate and predict multiple physical functions simultaneously. This unified, multi-task capability makes modeling complex, interdependent physical systems more robust, flexible, efficient, and practical.