{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,20]],"date-time":"2026-10-20T11:59:49Z","timestamp":1792497589993,"version":"4.5.0"},"reference-count":48,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Sci. Comput."],"published-print":{"date-parts":[[2025,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>The Fokker\u2013Planck (FP) equation is a foundational partial differential equation (PDE) in stochastic processes involving Brownian motions. However, the curse of dimensionality (CoD) poses a formidable challenge when dealing with high-dimensional FP equations. Although Monte Carlo simulation and (vanilla) Physics-Informed Neural Networks (PINNs) have shown the potential to tackle CoD, both methods exhibit significant numerical errors in high dimensions when dealing with the probability density function (PDF) associated with Brownian motion. The pointwise PDF values tend to decrease exponentially as dimensionality increases, surpassing the precision of numerical simulations and resulting in substantial errors. Moreover, due to its massive sampling, Monte Carlo fails to offer fast sampling. Modeling the logarithm likelihood (LL) via vanilla PINNs transforms the FP equation into a notoriously difficult Hamilton\u2013Jacobi\u2013Bellman (HJB) equation, which is impractical for PINN learning, whose error grows rapidly with dimension. To this end, we propose a novel approach utilizing a score-based solver to fit the score function in stochastic differential equations (SDEs). The score function, defined as the gradient of the LL, plays a fundamental role in inferring LL and PDF and enables fast SDE sampling, offering an effective means to overcome the CoD. Three fitting methods, Score Matching (SM), Sliced Score Matching (SSM), and Score-PINN, are introduced, each contributing unique advantages in computational complexity, accuracy, and generality. The proposed score-based SDE solver operates in two stages: first, employing SM or Score-PINN to acquire the score function; and second, solving the LL via an ordinary differential equation (ODE) using the obtained score function. Comparative evaluations across these methods showcase varying tradeoffs. The proposed methodology is evaluated across diverse SDEs, including anisotropic Ornstein\u2013Uhlenbeck processes, geometric Brownian motion, and Brownian motion with varying eigenspace. We also test various distributions, including Gaussian, Log-normal, Laplace, and Cauchy distributions. The numerical results demonstrate the score-based SDE solver\u2019s stability, speed, and performance across different experimental settings, solidifying its potential as a solution to CoD for high-dimensional FP equations. Our code is available at https:\/\/github.com\/zheyuanhu01\/Score_PINN .<\/jats:p>\n                  <jats:p>Reproducibility of computational results. This paper has been awarded the \u201cSIAM Reproducibility Badge: Code and data available\u201d as a recognition that the authors have followed reproducibility principles valued by SISC and the scientific computing community. Code and data that allow readers to reproduce the results in this paper are available at https:\/\/github.com\/zheyuanhu01\/Score_PINN and in the supplementary material ( Score_PINN-main.zip [17KB]). [Formula: see text]<\/jats:p>","DOI":"10.1137\/24m1638768","type":"journal-article","created":{"date-parts":[[2025,6,6]],"date-time":"2025-06-06T03:01:51Z","timestamp":1749178911000},"page":"C680-C705","source":"Crossref","is-referenced-by-count":21,"title":["Score-Based Physics-Informed Neural Networks for High-Dimensional Fokker\u2013Planck Equations"],"prefix":"10.1137","volume":"47","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0116-9874","authenticated-orcid":true,"given":"Zheyuan","family":"Hu","sequence":"first","affiliation":[{"name":"Department of Computer Science, National University of Singapore, Singapore, 119077."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8032-7510","authenticated-orcid":true,"given":"Zhongqiang","family":"Zhang","sequence":"additional","affiliation":[{"name":"Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9713-7120","authenticated-orcid":true,"given":"George E.","family":"Karniadakis","sequence":"additional","affiliation":[{"name":"Division of Applied Mathematics, Brown University, Providence, RI 02912 USA, and Advanced Computing, Mathematics and Data Division, Pacific Northwest National Laboratory, Richland, WA 99352 USA."}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kenji","family":"Kawaguchi","sequence":"additional","affiliation":[{"name":"Department of Computer Science, National University of Singapore, Singapore, 119077."}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2025,6,6]]},"reference":[{"key":"ref1","unstructured":"H. Alhussein, M. Khasawneh, and M. F. Daqaq, Physics-Informed Solution of the Stationary Fokker-Plank Equation for a Class of Nonlinear Dynamical Systems: An Evaluation Study, preprint, arXiv:2309.16725, 2023."},{"key":"ref2","doi-asserted-by":"publisher","DOI":"10.1137\/19M1297919"},{"key":"ref3","doi-asserted-by":"publisher","DOI":"10.1007\/s42985-024-00272-4"},{"key":"ref4","volume-title":"Dynamic Programming","author":"Bellman R.","year":"1957"},{"key":"ref5","doi-asserted-by":"publisher","DOI":"10.1088\/2632-2153\/ace2aa"},{"key":"ref6","doi-asserted-by":"crossref","unstructured":"N. M. Boffi and E. Vanden-Eijnden, Deep learning probability flows and entropy production rates in active matter, Proc. Natl. Acad. Sci. USA, 121 (2024), e2318106121, https:\/\/doi.org\/10.1073\/pnas.2318106121.","DOI":"10.1073\/pnas.2318106121"},{"key":"ref7","unstructured":"J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. VanderPlas, S. Wanderman-Milne, and Q. Zhang, JAX: Composable transformations of Python+NumPy programs, 2018, http:\/\/github.com\/google\/jax."},{"key":"ref8","first-page":"6572","volume-title":"Advances in Neural Information Processing Systems Curran Associates","volume":"31","author":"Chen R. T.","year":"2018"},{"key":"ref9","doi-asserted-by":"publisher","DOI":"10.1137\/20M1360153"},{"key":"ref10","doi-asserted-by":"crossref","first-page":"23761","DOI":"10.52202\/075280-1032","volume":"36","author":"Cho J.","year":"2023","journal-title":"Adv. Neural Inf. Process. Syst."},{"key":"ref11","doi-asserted-by":"publisher","DOI":"10.1137\/080714130"},{"key":"ref12","doi-asserted-by":"publisher","DOI":"10.1007\/s42985-021-00089-5"},{"key":"ref13","doi-asserted-by":"publisher","DOI":"10.4208\/cicp.OA-2022-0090"},{"key":"ref14","doi-asserted-by":"publisher","DOI":"10.1137\/21M1445363"},{"key":"ref15","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111202"},{"key":"ref16","doi-asserted-by":"publisher","DOI":"10.1137\/1004050"},{"key":"ref17","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.1718942115"},{"key":"ref18","unstructured":"D. He, S. Li, W. Shi, X. Gao, J. Zhang, J. Bian, L. Wang, and T.Y. Liu, Learning physics-informed neural networks without stacked back-propagation, in 26th International Conference on Artificial Intelligence and Statistics, PMLR 206, 2023, pp. 3034\u20133047."},{"key":"ref19","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2024.116883"},{"key":"ref20","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2024.106369"},{"key":"ref21","unstructured":"Z. Hu, Z. Yang, Y. Wang, G. E. Karniadakis, and K. Kawaguchi, Bias-Variance Trade-off in Physics-Informed Neural Networks with Randomized Smoothing for High-Dimensional PDEs, preprint, arXiv:2311.15283, 2023."},{"key":"ref22","first-page":"695","volume":"6","author":"Hyv\u00e4rinen A.","year":"2005","journal-title":"J. Mach. Learn. Res."},{"key":"ref23","unstructured":"D. P. Kingma and J. Ba, Adam: A method for stochastic gradient descent, in ICLR: International Conference on Learning Representations, ICLR US, 2015, pp. 1\u201315."},{"key":"ref24","doi-asserted-by":"publisher","DOI":"10.1007\/BF02716786"},{"key":"ref25","unstructured":"C.H. Lai, Y. Takida, N. Murata, T. Uesaka, Y. Mitsufuji, and S. Ermon, FP-diffusion: Improving score-based diffusion models by enforcing the underlying score Fokker-Planck equation, in International Conference on Machine Learning, PMLR, 2023, pp. 18365\u201318398."},{"key":"ref26","doi-asserted-by":"publisher","DOI":"10.1016\/j.amc.2023.128185"},{"key":"ref27","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-022-01844-5"},{"key":"ref28","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111783"},{"key":"ref29","doi-asserted-by":"publisher","DOI":"10.1137\/20M1344986"},{"key":"ref30","doi-asserted-by":"publisher","DOI":"10.1137\/21M1397908"},{"key":"ref31","doi-asserted-by":"publisher","DOI":"10.1063\/5.0065093"},{"key":"ref32","unstructured":"P. Mandal and A. Apte, Learning Zeros of Fokker-Planck Operators, preprint, arXiv:2306.07068, 2023."},{"key":"ref33","volume-title":"Stochastic Differential Equations: An Introduction with Applications","author":"Oksendal B.","year":"2013"},{"key":"ref34","first-page":"19175","volume-title":"in Advances in Neural Information Processing Systems 33","author":"Pang T.","year":"2020"},{"key":"ref35","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2023.112464"},{"key":"ref36","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-007-5134-7_5"},{"key":"ref37","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2018.10.045"},{"key":"ref38","first-page":"187","volume":"262","author":"Sepehrian B.","year":"2015","journal-title":"Appl. Math. Comput."},{"key":"ref39","unstructured":"Y. Song, S. Garg, J. Shi, and S. Ermon, Sliced score matching: A scalable approach to density and score estimation, in Proceedings of the Thirty-Fifth Conference on Uncertainty in Artificial Intelligence, UAI 2019, Tel Aviv, Israel, 2019, 204, http:\/\/auai.org\/uai2019\/proceedings\/papers\/204.pdf."},{"key":"ref40","unstructured":"Y. Song, J. Sohl-Dickstein, D. P. Kingma, A. Kumar, S. Ermon, and B. Poole, Score-based generative modeling through stochastic differential equations, in International Conference on Learning Representations, 2021, https:\/\/openreview.net\/forum?id=PxTIG12RRHS."},{"key":"ref41","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2022.111080"},{"key":"ref42","volume-title":"Advances in Neural Information Processing Systems","author":"Wang C.","year":"2022"},{"key":"ref43","doi-asserted-by":"publisher","DOI":"10.1016\/j.neunet.2025.107165"},{"key":"ref44","doi-asserted-by":"crossref","unstructured":"Y. Wang, H. Xie, and P. Jin, Tensor neural network and its numerical integration, 42 (2024), pp. 1714\u20131742.","DOI":"10.4208\/jcm.2307-m2022-0233"},{"key":"ref45","doi-asserted-by":"publisher","DOI":"10.1016\/0021-9991(70)90044-6"},{"key":"ref46","unstructured":"J. Zhai, M. Dobson, and Y. Li, A deep learning method for solving Fokker-Planck equations, in Proceedings of the 2nd Mathematical and Scientific Machine Learning, PMLR 145, 2022, pp. 568\u2013597."},{"key":"ref47","doi-asserted-by":"publisher","DOI":"10.1007\/s11071-022-07361-2"},{"key":"ref48","unstructured":"Y. Zhao, X. Yu, Z. Chen, Z. Liu, S. Liu, and Z. Zhang, Tensor-Compressed Back-Propagation-Free Training for (Physics-Informed) Neural Networks, preprint, arXiv:2308.09858, 2023."}],"container-title":["SIAM Journal on Scientific Computing"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/epubs.siam.org\/doi\/pdf\/10.1137\/24M1638768","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T15:08:44Z","timestamp":1787324924000},"score":1,"resource":{"primary":{"URL":"https:\/\/epubs.siam.org\/doi\/10.1137\/24M1638768"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,6]]},"references-count":48,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2025,6,30]]}},"alternative-id":["10.1137\/24M1638768"],"URL":"https:\/\/doi.org\/10.1137\/24m1638768","relation":{},"ISSN":["1064-8275","1095-7197"],"issn-type":[{"value":"1064-8275","type":"print"},{"value":"1095-7197","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,6]]}}}