Chen Wang

Xiangshu Gong, Zhiqiang Xie, Xiaowei Jin et al.

Many problems are governed by differential equations (DEs). Artificial intelligence (AI) is a new path for solving DEs. However, data is very scarce and existing AI solvers struggle with approximation of high frequency components (AHFC). We propose an AI paradigm for solving diverse DEs, including DE-ruled first-principles data generation methodology and scale-dilation operator (SDO) AI solver. Using either prior knowledge or random fields, we generate solutions and then substitute them into the DEs to derive the sources and initial/boundary conditions through balancing DEs, thus producing arbitrarily vast amount of, first-principles-consistent training datasets at extremely low computational cost. We introduce a reversible SDO that leverages the Fourier transform of the multiscale solutions to fix AHFC, and design a spatiotemporally coupled, attention-based Transformer AI solver of DEs with SDO. An upper bound on the Hessian condition number of the loss function is proven to be proportional to the squared 2-norm of the solution gradient, revealing that SDO yields a smoother loss landscape, consequently fixing AHFC with efficient training. Extensive tests on diverse DEs demonstrate that our AI paradigm achieves consistently superior accuracy over state-of-the-art methods. This work makes AI solver of DEs to be truly usable in broad nature and engineering fields.

Zhanpeng Wang, Shenghao Li, Chen Wang et al.

In recent years, the knowledge surrounding diffusion models(DMs) has grown significantly, though several theoretical gaps remain. Particularly noteworthy is prior error, defined as the discrepancy between the termination distribution of the forward process and the initial distribution of the reverse process. To address these deficiencies, this paper explores the deeper relationship between optimal transport(OT) theory and DMs with discrete initial distribution. Specifically, we demonstrate that the two stages of DMs fundamentally involve computing time-dependent OT. However, unavoidable prior error result in deviation during the reverse process under quadratic transport cost. By proving that as the diffusion termination time increases, the probability flow exponentially converges to the gradient of the solution to the classical Monge-Ampère equation, we establish a vital link between these fields. Therefore, static OT emerges as the most intrinsic single-step method for bridging this theoretical potential gap. Additionally, we apply these insights to accelerate sampling in both unconditional and conditional generation scenarios. Experimental results across multiple image datasets validate the effectiveness of our approach.