In scenarios with limited available data, training the function-to-function neural PDE solver in an unsupervised manner is essential. However, the efficiency and accuracy of existing methods are constrained by the properties of numerical algorithms, such as finite difference and pseudo-spectral methods, integrated during the training stage. These methods necessitate careful spatiotemporal discretization to achieve reasonable accuracy, leading to significant computational challenges and inaccurate simulations, particularly in cases with substantial spatiotemporal variations. To address these limitations, we propose theMonte Carlo Neural PDE Solver (MCNP Solver) for training unsupervised neural solvers via the PDEs’ probabilistic representation, which regards macroscopic phenomena as ensembles of random particles. Compared to other unsupervised methods, MCNP Solver naturally inherits the advantages of the Monte Carlo method, which is robust against spatiotemporal variations and can tolerate coarse step size. In simulating the trajectories of particles, we employ Heun’smethod for the convection process and calculate the expectation via the probability density function of neighbouring grid points during the diffusion process. These techniques enhance accuracy and circumvent the computational issues associated with Monte Carlo sampling. Our numerical experiments on convection-diffusion, Allen-Cahn, and Navier-Stokes equations demonstrate significant improvements in accuracy and efficiency compared to other unsupervised baselines.
Publication:IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, VOL. 47, NO. 6, JUNE 2025 http://dx.doi.org/10.1109/TPAMI.2025.3548673
Author:Rui Zhang, Gaoling School of Artificial Intelligence, Renmin University of China, Beijing 100872, China
e-mail: rayzhang@ruc.edu.cn
Qi Meng, Shihua Zhang, and Zhi-Ming Ma, Academy of Mathematics and Systems Science, Chinese Academy of Sciences (CAS), Beijing
100190, China
e-mail: meq@amss.ac.cn; zsh@amss.ac.cn; mazm@amt.ac.cn
Rongchan Zhu, Beijing Institute of Technology, Beijing 100081,China
e-mail: zhurongchan@126.com
YueWang, Beijing Jiaotong University, Beijing 100044, China
e-mail:yuewang@gmail.com
Wenlei Shi, Institute of Computing Technology, Chinese Academy of Sciences, Beijing 100190, China
e-mail: shiwenlei22b@ict.ac.cn
Tie-Yan Liu, Zhongguancun Academy, Beijing 100094, China
e-mail:tie-yan.liu@outlook.com
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