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Our algorithm is meshfree, which is\nkey since meshes become infeasible in higher dimensions. Instead of forming a\nmesh, the neural network is trained on batches of randomly sampled time and\nspace points. The algorithm is tested on a class of high-dimensional free\nboundary PDEs, which we are able to accurately solve in up to $200$ dimensions.\nThe algorithm is also tested on a high-dimensional Hamilton-Jacobi-Bellman PDE\nand Burgers' equation. The deep learning algorithm approximates the general\nsolution to the Burgers' equation for a continuum of different boundary\nconditions and physical conditions (which can be viewed as a high-dimensional\nspace). We call the algorithm a \"Deep Galerkin Method (DGM)\" since it is\nsimilar in spirit to Galerkin methods, with the solution approximated by a\nneural network instead of a linear combination of basis functions. In addition,\nwe prove a theorem regarding the approximation power of neural networks for a\nclass of quasilinear parabolic PDEs.","url_abs":"http://arxiv.org/abs/1708.07469v5","url_pdf":"http://arxiv.org/pdf/1708.07469v5.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":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/ArturGoldman/NN_DE_solver","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/Plemeur/DGM","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/Plemeur/pre_stage","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"none","reach":{"status":"unanswered"}},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/alialaradi/DeepGalerkinMethod","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"unanswered"}},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/atapritchard/DPDEs","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"tf","reach":{"status":"unanswered"}},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/junbinhuang/DeepRitz","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"MIT"}},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/mjhwright/error-correction","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":null},{"paper_slug":"dgm-a-deep-learning-algorithm-for-solving","repo_url":"https://github.com/pooyasf/DGM","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"unanswered"}}],"tasks":[{"task_slug":"deep-learning","task_name":"Deep Learning"}],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=1708.07469","mcp":{"get_harvested_code_for_paper":{"arxiv_id":"1708.07469"}},"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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