This is an automated message from the <a href=\"https://huggingface.co/librarian-bots\">Librarian Bot</a>. I found the following papers similar to this paper. </p>\n<p>The following papers were recommended by the Semantic Scholar API </p>\n<ul>\n<li><a href=\"https://huggingface.co/papers/2605.07116\">Stabilized neural Hamilton--Jacobi--Bellman solvers: Error analysis and applications in model-based reinforcement learning</a> (2026)</li>\n<li><a href=\"https://huggingface.co/papers/2604.07762\">Generative optimal transport via forward-backward HJB matching</a> (2026)</li>\n<li><a href=\"https://huggingface.co/papers/2605.18566\">HJ-Gauss: A Monte-Carlo HJ Reachability Scheme</a> (2026)</li>\n<li><a href=\"https://huggingface.co/papers/2604.10191\">Policy Iteration for Stationary Discounted Hamilton--Jacobi--Bellman Equations: A Viscosity Approach</a> (2026)</li>\n<li><a href=\"https://huggingface.co/papers/2605.07157\">Learned Lagrangian Models of PDEs via Euler-Lagrange Residual Minimization</a> (2026)</li>\n<li><a href=\"https://huggingface.co/papers/2604.11829\">Learning on the Temporal Tangent Bundle for Physics-Informed Neural Networks</a> (2026)</li>\n<li><a href=\"https://huggingface.co/papers/2605.09058\">Nonlinear GENERIC Informed Neural Networks (N-GINNs): learning GENERIC dynamics with non-quadratic dissipation potentials</a> (2026)</li>\n</ul>\n<p> Please give a thumbs up to this comment if you found it helpful!</p>\n<p> If you want recommendations for any Paper on Hugging Face checkout <a href=\"https://huggingface.co/spaces/librarian-bots/recommend_similar_papers\">this</a> Space</p>\n<p> You can directly ask Librarian Bot for paper recommendations by tagging it in a comment: <code><span class=\"SVELTE_PARTIAL_HYDRATER contents\" data-target=\"UserMention\" data-props=\"{&quot;user&quot;:&quot;librarian-bot&quot;}\"><span class=\"inline-block\"><span class=\"contents\"><a href=\"/librarian-bot\">@<span class=\"underline\">librarian-bot</span></a></span> </span></span> recommend</code></p>\n","updatedAt":"2026-06-03T01:59:03.045Z","author":{"_id":"63d3e0e8ff1384ce6c5dd17d","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/1674830754237-63d3e0e8ff1384ce6c5dd17d.jpeg","fullname":"Librarian Bot (Bot)","name":"librarian-bot","type":"user","isPro":false,"isHf":false,"isHfAdmin":false,"isMod":false,"followerCount":360,"isUserFollowing":false}},"numEdits":0,"identifiedLanguage":{"language":"en","probability":0.7347700595855713},"editors":["librarian-bot"],"editorAvatarUrls":["https://cdn-avatars.huggingface.co/v1/production/uploads/1674830754237-63d3e0e8ff1384ce6c5dd17d.jpeg"],"reactions":[],"isReport":false}}],"primaryEmailConfirmed":false,"paper":{"id":"2605.28983","authors":[{"_id":"6a1ea90e808ddbc3c7d43ff3","name":"Jose Marie Antonio Miñoza","hidden":false},{"_id":"6a1ea90e808ddbc3c7d43ff4","name":"Erika Fille T. Legara","hidden":false},{"_id":"6a1ea90e808ddbc3c7d43ff5","name":"Christopher P. Monterola","hidden":false}],"publishedAt":"2026-05-27T00:00:00.000Z","submittedOnDailyAt":"2026-06-02T00:00:00.000Z","title":"The Hamilton-Jacobi Theory of Deep Learning","submittedOnDailyBy":{"_id":"647629b2a4a46a7bba1b70c3","avatarUrl":"/avatars/bf87597b02319f31651447e1123a1fec.svg","isPro":false,"fullname":"Jose Marie Antonio Minoza","user":"jomaminoza","type":"user","name":"jomaminoza"},"summary":"In this paper, training a neural network is identified, exactly, as a search through Hamilton--Jacobi initial-value problems: each gradient step selects the initial data of a viscous Hamilton--Jacobi equation whose Hopf--Cole propagator best fits the observations; at inference, the input is the spatial point at which that solution is evaluated and the initial condition is already encoded in the weights. The correspondence is exact for log-sum-exp layers and structural for broader architectures: residual networks, transformers, and recurrent architectures (RNNs, LSTMs, SSMs) each discretize the same class of Hamilton--Jacobi equations, with architecture-dependent Hamiltonian and viscosity. A single deformation parameter varepsilon unifies all four perspectives (network, tropical algebra, viscous PDE, convex optimization) in a commutative diagram closed under Lipschitz conditions. Quantitative consequences include: the minimax optimal generalization rate O(n^{-1/(d+2)}) for fixed t; adversarial robustness controlled by varepsilon; backpropagation as the co-state equation of the Hamiltonian system for residual networks (Pontryagin Maximum Principle); scaling exponents consistent with data intrinsic dimension via PDE quadrature; and a closed-form O(N) influence function (softmax attribution weights π_j) whose entropy landscape undergoes fold bifurcations as varepsilon increases, each merging attribution basins.","upvotes":1,"discussionId":"6a1ea90e808ddbc3c7d43ff6","ai_summary":"Neural network training is formulated as a Hamilton--Jacobi initial-value problem where gradient steps correspond to solving viscous Hamilton--Jacobi equations, with connections to residual networks, transformers, and RNNs through shared mathematical structures.","ai_keywords":["Hamilton--Jacobi initial-value problems","viscous Hamilton--Jacobi equation","Hopf--Cole propagator","log-sum-exp layers","residual networks","transformers","recurrent architectures","RNNs","LSTMs","SSMs","tropical algebra","convex optimization","minimax optimal generalization rate","adversarial robustness","backpropagation","co-state equation","Pontryagin Maximum Principle","PDE quadrature","influence function","softmax attribution weights","entropy landscape","fold bifurcations"],"ai_summary_model":"Qwen/Qwen2.5-Coder-32B-Instruct","organization":{"_id":"673dfcb072711222f01e29fd","name":"cair-ph","fullname":"Center for AI Research","avatar":"https://cdn-avatars.huggingface.co/v1/production/uploads/647629b2a4a46a7bba1b70c3/-cCnBfB_4QNfFt6xqEtv9.png"}},"canReadDatabase":false,"canManagePapers":false,"canSubmit":false,"hasHfLevelAccess":false,"upvoted":false,"upvoters":[{"_id":"69ccc93092e44910c0000aef","avatarUrl":"https://cdn-avatars.huggingface.co/v1/production/uploads/noauth/mi2JPBKB8k5sCqQ2jP4bh.png","isPro":false,"fullname":"James Martin","user":"hansh6","type":"user"}],"acceptLanguages":["en"],"dailyPaperRank":0,"organization":{"_id":"673dfcb072711222f01e29fd","name":"cair-ph","fullname":"Center for AI Research","avatar":"https://cdn-avatars.huggingface.co/v1/production/uploads/647629b2a4a46a7bba1b70c3/-cCnBfB_4QNfFt6xqEtv9.png"},"markdownContentUrl":"https://huggingface.co/buckets/huggingchat/papers-content/resolve/2605/2605.28983.md"}">

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