Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation

doi:10.1109/TPAMI.2025.3548673

	Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation
	Zhang, Rui 1; Meng, Qi 2; Zhu, Rongchan 3; Wang, Yue 4; Shi, Wenlei 5; Zhang, Shihua 2; Ma, Zhi-Ming 2; Liu, Tie-Yan 6
	2025-06-01
发表期刊	IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE
ISSN	0162-8828
卷号	47 期号:6 页码:5059-5075
摘要	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 the Monte 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's method 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.
关键词	Training Monte Carlo methods Spatiotemporal phenomena Accuracy Probabilistic logic Neural networks Numerical models Mathematical models Finite difference methods Diffusion processes Neural PDE solver Monte Carlo method Feynman-Kac formula AI for PDE
DOI	10.1109/TPAMI.2025.3548673
收录类别	SCI
语种	英语
资助项目	NSFC[9247010235] ; National Key Research and Development Program of China[2019YFA0709501] ; CAS Project for Young Scientists in Basic Research[YSBR-034]
WOS研究方向	Computer Science ; Engineering
WOS类目	Computer Science, Artificial Intelligence ; Engineering, Electrical & Electronic
WOS记录号	WOS:001484716600041
出版者	IEEE COMPUTER SOC
引用统计
文献类型	期刊论文
条目标识符	http://119.78.100.204/handle/2XEOYT63/42380
专题	中国科学院计算技术研究所期刊论文_英文
通讯作者	Meng, Qi
作者单位	1.Renmin Univ China, Gaoling Sch Artificial Intelligence, Beijing 100872, Peoples R China 2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China 3.Beijing Inst Technol, Beijing 100081, Peoples R China 4.Beijing Jiaotong Univ, Beijing 100044, Peoples R China 5.Chinese Acad Sci, Inst Comp Technol, Beijing 100190, Peoples R China 6.Zhongguancun Acad, Beijing 100094, Peoples R China
推荐引用方式 GB/T 7714	Zhang, Rui,Meng, Qi,Zhu, Rongchan,et al. Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation[J]. IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,2025,47(6):5059-5075.
APA	Zhang, Rui.,Meng, Qi.,Zhu, Rongchan.,Wang, Yue.,Shi, Wenlei.,...&Liu, Tie-Yan.(2025).Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation.IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE,47(6),5059-5075.
MLA	Zhang, Rui,et al."Monte Carlo Neural PDE Solver for Learning PDEs via Probabilistic Representation".IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE 47.6(2025):5059-5075.