{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/physics-constrained-deep-learning-for-high","title":"Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data","arxiv_id":"1901.06314","date":"2019-01-18","proceeding":null,"authors":["Yinhao Zhu","Nicholas Zabaras","Phaedon-Stelios Koutsourelakis","Paris Perdikaris"],"abstract":"Surrogate modeling and uncertainty quantification tasks for PDE systems are\nmost often considered as supervised learning problems where input and output\ndata pairs are used for training. The construction of such emulators is by\ndefinition a small data problem which poses challenges to deep learning\napproaches that have been developed to operate in the big data regime. Even in\ncases where such models have been shown to have good predictive capability in\nhigh dimensions, they fail to address constraints in the data implied by the\nPDE model. This paper provides a methodology that incorporates the governing\nequations of the physical model in the loss/likelihood functions. The resulting\nphysics-constrained, deep learning models are trained without any labeled data\n(e.g. employing only input data) and provide comparable predictive responses\nwith data-driven models while obeying the constraints of the problem at hand.\nThis work employs a convolutional encoder-decoder neural network approach as\nwell as a conditional flow-based generative model for the solution of PDEs,\nsurrogate model construction, and uncertainty quantification tasks. The\nmethodology is posed as a minimization problem of the reverse Kullback-Leibler\n(KL) divergence between the model predictive density and the reference\nconditional density, where the later is defined as the Boltzmann-Gibbs\ndistribution at a given inverse temperature with the underlying potential\nrelating to the PDE system of interest. The generalization capability of these\nmodels to out-of-distribution input is considered. Quantification and\ninterpretation of the predictive uncertainty is provided for a number of\nproblems.","url_abs":"http://arxiv.org/abs/1901.06314v1","url_pdf":"http://arxiv.org/pdf/1901.06314v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"abstracts"},"code_links":[{"paper_slug":"physics-constrained-deep-learning-for-high","repo_url":"https://github.com/cics-nd/pde-surrogate","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}}],"tasks":[{"task_slug":"small-data","task_name":"Small Data Image Classification"},{"task_slug":"uncertainty-quantification","task_name":"Uncertainty Quantification"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1901.06314","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1901.06314"}},"developers":"https://syntology.ai/developers","read_at":"2026-09-24T18:15:14+00:00","read_at_is":"when the build read Syntology's graph, not when any sample ran","claim":"Per-sample execution status on synthesized fixtures; not a correctness claim about the paper. 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