{"paper":{"title":"Neural Ordinary Differential Equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"Deep neural networks can replace discrete layers with continuous dynamics defined by ordinary differential equations.","cross_cats":["cs.AI","stat.ML"],"primary_cat":"cs.LG","authors_text":"David Duvenaud, Jesse Bettencourt, Ricky T. Q. Chen, Yulia Rubanova","submitted_at":"2018-06-19T17:50:12Z","abstract_excerpt":"We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"That a neural network can be trained to produce a vector field whose integral yields useful representations, and that standard ODE solvers remain numerically stable and differentiable enough for end-to-end gradient descent across the range of problems considered.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Neural networks are redefined as continuous dynamical systems by learning the derivative of the hidden state with a neural network and integrating it with an ODE solver.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"Deep neural networks can replace discrete layers with continuous dynamics defined by ordinary differential equations.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"abc96746c8234081dcaad4738c14dd9557ecf382d7526a05e789a62840b98134"},"source":{"id":"1806.07366","kind":"arxiv","version":5},"verdict":{"id":"6da4881e-7ef9-4968-839f-2b0efd2c0cfd","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T12:56:04.870070Z","strongest_claim":"We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver.","one_line_summary":"Neural networks are redefined as continuous dynamical systems by learning the derivative of the hidden state with a neural network and integrating it with an ODE solver.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"That a neural network can be trained to produce a vector field whose integral yields useful representations, and that standard ODE solvers remain numerically stable and differentiable enough for end-to-end gradient descent across the range of problems considered.","pith_extraction_headline":"Deep neural networks can replace discrete layers with continuous dynamics defined by ordinary differential equations."},"references":{"count":60,"sample":[{"doi":"","year":2011,"title":"Computationally efficient convolved multiple output G aussian processes","work_id":"a79ccadd-e746-419c-8444-fcba767fd3b6","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2017,"title":"OptNet : Differentiable optimization as a layer in neural networks","work_id":"3b7f8d99-7b4a-4bdb-9e76-305ba58e9a5b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2013,"title":"A general-purpose software framework for dynamic optimization","work_id":"f244a44e-8889-43b2-be19-547832ef321b","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"CasADi -- A software framework for nonlinear optimization and optimal control","work_id":"c5e1095a-3dfa-41f5-a35c-16af5c5a3219","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Automatic differentiation in machine learning: a survey","work_id":"8abd3335-a598-4a1c-82f1-435c37825259","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":60,"snapshot_sha256":"81e54cc4e9d8e00e01936c37c564cb69b86b298f26e3b1411a1aa0b3d412b5df","internal_anchors":16},"formal_canon":{"evidence_count":3,"snapshot_sha256":"4941d4f57c642bb9ff2b1966790074421c88e13c5b66503d45885f99b68d26c8"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}