Xun Yang

Xun Yang, Guanqiu Ma, Maohua Ran

When solving time-dependent partial differential equations(PDEs), traditional physics-informed neural networks (PINNs) have inherent limitations: due to the lack of temporal causality, the network is forced to learn the later-time control equations while fully capturing the initial conditions, resulting in the continuous accumulation of errors during the integration process. Meanwhile, the limited expressivity of a single network hinders its ability to capture diverse physical behaviors across multiple subdomains. To address these issues, we propose a domain-decomposition-based causal PINNs (DDC-PINNs) framework. This framework enhances spatial representation through domain decomposition and employs a causal strategy to constrain the temporal learning sequence, thereby improving the accuracy and generalization ability of solutions to time-varying problems. Within this framework, an approximate solution is first obtained through PINNs with domain decomposition. Subsequently, the time derivative term in the PDE is retained, while other solution-dependent terms are replaced with this approximate solution, thereby simplifying the original PDEs into ordinary differential equations (ODEs). Finally, classical numerical methods for solving ODEs are employed to obtain the time-dependent solution. DDC-PINNs not only preserve the inherent computational efficiency and flexibility of PINNs but also effectively incorporate causality when solving time-dependent PDEs. Numerical experiments verify the effectiveness of the proposed method.

Ruoyu Wang, Ziyu Li, Beier Zhu et al.

Diffusion models (DMs) have achieved state-of-the-art generative performance but suffer from high sampling latency due to their sequential denoising nature. Existing solver-based acceleration methods often face significant image quality degradation under a low-latency budget, primarily due to accumulated truncation errors arising from the inability to capture high-curvature trajectory segments. In this paper, we propose the Ensemble Parallel Direction solver (dubbed as EPD-Solver), a novel ODE solver that mitigates these errors by incorporating multiple parallel gradient evaluations in each step. Motivated by the geometric insight that sampling trajectories are largely confined to a low-dimensional manifold, EPD-Solver leverages the Mean Value Theorem for vector-valued functions to approximate the integral solution more accurately. Importantly, since the additional gradient computations are independent, they can be fully parallelized, preserving low-latency sampling nature. We introduce a two-stage optimization framework. Initially, EPD-Solver optimizes a small set of learnable parameters via a distillation-based approach. We further propose a parameter-efficient Reinforcement Learning (RL) fine-tuning scheme that reformulates the solver as a stochastic Dirichlet policy. Unlike traditional methods that fine-tune the massive backbone, our RL approach operates strictly within the low-dimensional solver space, effectively mitigating reward hacking while enhancing performance in complex text-to-image (T2I) generation tasks. In addition, our method is flexible and can serve as a plugin (EPD-Plugin) to improve existing ODE samplers.