%0 Generic %1 rackauckas2020universal %A Rackauckas, Christopher %A Ma, Yingbo %A Martensen, Julius %A Warner, Collin %A Zubov, Kirill %A Supekar, Rohit %A Skinner, Dominic %A Ramadhan, Ali %A Edelman, Alan %D 2020 %K deeplearning neuralpde pde ude %T Universal Differential Equations for Scientific Machine Learning %U http://arxiv.org/abs/2001.04385 %X In the context of science, the well-known adage ä picture is worth a thousand words" might well be ä model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.

@misc{rackauckas2020universal, abstract = {In the context of science, the well-known adage "a picture is worth a thousand words" might well be "a model is worth a thousand datasets." Scientific models, such as Newtonian physics or biological gene regulatory networks, are human-driven simplifications of complex phenomena that serve as surrogates for the countless experiments that validated the models. Recently, machine learning has been able to overcome the inaccuracies of approximate modeling by directly learning the entire set of nonlinear interactions from data. However, without any predetermined structure from the scientific basis behind the problem, machine learning approaches are flexible but data-expensive, requiring large databases of homogeneous labeled training data. A central challenge is reconciling data that is at odds with simplified models without requiring "big data". In this work we develop a new methodology, universal differential equations (UDEs), which augments scientific models with machine-learnable structures for scientifically-based learning. We show how UDEs can be utilized to discover previously unknown governing equations, accurately extrapolate beyond the original data, and accelerate model simulation, all in a time and data-efficient manner. This advance is coupled with open-source software that allows for training UDEs which incorporate physical constraints, delayed interactions, implicitly-defined events, and intrinsic stochasticity in the model. Our examples show how a diverse set of computationally-difficult modeling issues across scientific disciplines, from automatically discovering biological mechanisms to accelerating the training of physics-informed neural networks and large-eddy simulations, can all be transformed into UDE training problems that are efficiently solved by a single software methodology.}, added-at = {2021-09-21T09:20:57.000+0200}, author = {Rackauckas, Christopher and Ma, Yingbo and Martensen, Julius and Warner, Collin and Zubov, Kirill and Supekar, Rohit and Skinner, Dominic and Ramadhan, Ali and Edelman, Alan}, biburl = {https://www.bibsonomy.org/bibtex/22b732149c0607358f29a27f672cca68b/annakrause}, description = {2001.04385.pdf}, interhash = {e27fc1cb469bf257cc30f74b0d556760}, intrahash = {2b732149c0607358f29a27f672cca68b}, keywords = {deeplearning neuralpde pde ude}, note = {cite arxiv:2001.04385Comment: 3 figures, 2 tables, 3 supplemental figures, 27 pages, 18 supplemental pages}, timestamp = {2021-09-21T09:20:57.000+0200}, title = {Universal Differential Equations for Scientific Machine Learning}, url = {http://arxiv.org/abs/2001.04385}, year = 2020 }