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 (ACMFD), a versatile probabilistic surrogate model for multi-physics emulation. ACMFD 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 then introduce an innovative denoising loss. The training involves randomly sampling the conditioned part and fitting the corresponding predicted noise to zero, enabling ACMFD to flexibly generate function values conditioned on any other functions or quantities. To enable efficient training and sampling, and to flexibly handle irregularly sampled data, we use GPs to interpolate function samples onto a grid, inducing a Kronecker product structure for efficient computation. We demonstrate the advantages of ACMFD across several fundamental multi-physics systems.

翻译：现代物理仿真通常涉及多个关注函数，而传统数值方法以复杂和计算成本高昂著称。虽然基于机器学习的代理模型能够显著降低成本，但多数模型仅专注于单一任务（如前向预测），且通常缺乏不确定性量化——这在许多应用中是不可或缺的组成部分。为突破这些限制，我们提出任意条件多功能扩散模型，这是一种用于多物理场仿真的通用概率代理模型。该模型能在单一框架内执行广泛任务，包括前向预测、各类反问题求解，以及基于其他量条件约束下对整个系统或子集量的数据仿真。具体而言，我们通过将噪声建模为高斯过程，扩展了标准去噪扩散概率模型以实现多功能生成。随后我们引入了一种创新的去噪损失函数。训练过程中随机采样条件约束部分，并将对应的预测噪声拟合至零，使模型能够灵活生成以任意其他函数或量为条件的函数值。为实现高效训练与采样，并灵活处理非均匀采样数据，我们采用高斯过程将函数样本插值至网格，通过诱导克罗内克积结构实现高效计算。我们在多个基础多物理场系统中验证了该模型的优势。