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One of the open problems in scientific computing is the long-time integration of nonlinear stochastic partial differential equations (SPDEs).
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T. P. Sapsis, Dynamically orthogonal field equations for stochastic fluid flows and particle dynamics, Ph.D. thesis, MIT, 2011
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Numerical schemes for dynamically orthogonal equations of stochastic fluid and ocean flows,
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Physics informed deep learning (part II): Data-driven discovery of nonlinear partial differential equations,
M. Raissi, P. Perdikaris, G. E. Karniadakis, · 2017
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A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems,
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Physics informed deep learning (part I): Data-driven solutions of nonlinear partial differential equations,
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Hidden physics models: Machine learning of nonlinear partial differential equations,
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M. Choi, Time-dependent Karhunen-Loève type decomposition methods for SPDEs, Ph.D. thesis, Brown University, 2014
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Stochastic collocation on unstructured multivariate meshes,
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Error analysis of the dynamically orthogonal approximation of time dependent random pdes,
E. Musharbash, F. Nobile, T. Zhou, · 2015
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Energy-optimal path planning by stochastic dynamically orthogonal level-set optimization,
D. Subramani, P. F. J. Lermusiaux, · 2016
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A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms,
M. Cheng, T. Y. Hou, Z. Zhang,
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A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: Adaptivity and generalizations,
M. Cheng, T. Y. Hou, Z. Zhang,
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Forward-backward stochastic neural networks: Deep learning of high-dimensional partial differential equations,
M. Raissi, · 2018
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Hidden fluid mechanics: Navier-Stokes informed deep learning from the passive scalar transport,
A. Yazdani, M. Raissi, G. E. Karniadakis, · 2018
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Deep learning of vortex-induced vibrations,
M. Raissi, Z. Wang, M. S. Triantafyllou, G. E. Karniadakis, · 2019
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Quantifying total uncertainty in physics-informed neural networks for solving forward and inverse stochastic problems,
D. Zhang, L. Lu, L. Guo, G. E. Karniadakis, · 2019
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