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La Latent-sp space Dynam Dynamics ics for r Re Reduced Deformable Simulation Lawson Fulton 1,2 , Vismay Modi 1 , David Duvenaud 1 , David I.W. Levin 1 , Alec Jacobson 1 1 University of Toronto, Canada 2 MESH Consultants, Canada The 40

  1. La Latent-sp space Dynam Dynamics ics for r Re Reduced Deformable Simulation Lawson Fulton 1,2 , Vismay Modi 1 , David Duvenaud 1 , David I.W. Levin 1 , Alec Jacobson 1 1 University of Toronto, Canada 2 MESH Consultants, Canada The 40° Annual Conference of the European Association for Computer Graphics

  2. Wh Why y de deformabl ble si simul ulation? n? [Ziva Dynamics]

  3. Re Research Question Can we use machine learning to accelerate hyperelastic simulation?

  4. Re Related Work Latent-space Physics: Towards Learning the Temporal Evolution of Fluid Flow Deep Fluids – A Generative Network for Wiewel et al. 2019 Parameterized Fluid Simulations Kim et al. 2019 Learn how to update the latent state of a system

  5. Re Related Work DeepWarp: DNN-based Nonlinear Deformation Luo et al. 2018 Neural Material: Learning Elastic Constitutive Material and Damping Models from Sparse Data Wang et al. 2018 Learn correction to cheap simulation

  6. Ou Our Approach Build on the vast literature of Model Reduction Simulate in nonlinear latent space using the true equations of motion

  7. <latexit sha1_base64="FEpvwDVn/P/rPOKwRTc2i8k/Bio=">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</latexit> <latexit sha1_base64="v2GJEcFuALMEXUHiFfRpE01uSbU=">ACG3icbZDLSsNAFIYn9VbrerSzWAR6qYkadF2IRQFcSNUsBdoSplMJu3QyYWZiVBC3sONr+LGhSKuBe+jUmaej8w8PH/53DO/KbPqJCq+q7kFhaXlfyq4W19Y3NreL2Tkd4AcekjT3m8Z6JBGHUJW1JSM9nxPkmIx0zclZ4ndvCBfUc6/l1CcDB41calOMZCwNi7rhIDk27fA8Ks8xiA5P5nwZGZblyfDLi4bFklpR04J/QcugBLJqDYuvhuXhwCGuxAwJ0dUXw5CxCXFjEQFIxDER3iCRqQfo4scIgZh+rcIHsSKBW2Px8+VMFW/T4TIEWLqmHFncqL47SXif14/kHZ9EFLXDyRx8WyRHTAoPZgEBS3KCZsGgPCnMa3QjxGHGEZx1lIQ2jUNb1egynUjuoz0KuNzxA6ekWrVvSrWql5msWRB3tgH5SBo5BE1yAFmgDG7BPXgET8qd8qA8Ky+z1pySzeyCH6W8fQBWvaMm</latexit> <latexit sha1_base64="1sCQ9Z9V+VFKHwLQ+uRbCAt8iFE=">AB+HicbZBLS8NAFIUn9VXro1GXbgaL4ELKpA2a7opuXFawD2hDmUwn7dDJMxMxBr6S9y4UMStP8Wd/8Y0DeLrwMDHOfcyl+NFnCmN0IdRWFldW98obpa2tnd2y+befkeFsS0TUIeyp6HFeVM0LZmtNeJCkOPE673vRykXdvqVQsFDd6FlE3wGPBfEawTq2hWRYDHEUyvIM2OkUIDc0KqJM8C9YOVRArtbQfB+MQhIHVGjCsVJ9C0XaTbDUjHA6Lw1iRSNMpnhM+ykKHFDlJtnhc3icOiPohzJ9QsPM/b6R4ECpWeClkwHWE/U7W5j/Zf1Y+46bMBHFmgqy/MiPOdQhXLQAR0xSovksBUwkS2+FZIlJjrtqpSV0HCsmPDOwzZwm1euOrhE6tatWrtWu70rzI6yiCQ3AEToAFzkETXIEWaAMCYvAnsCzcW8Gi/G63K0YOQ7B+CHjLdPB/2Sg=</latexit> Fir First, t, wh why is is it it slo low? w?   n ≈ 40 , 000 p 1 p 2     u = .   . .   p n Vertex Displacements Solving large differential equation F ( u ) = M ¨ u

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  9. <latexit sha1_base64="UtsE7jvHuKmjUwWZfT06HGBy+I=">ACnicbZDLSsNAFIYnXmu9RV26GS2Cq5KkRdOFUHTjskLTFtpYJtNJO3RycWYilJC1G1/FjQtF3PoE7nwb0zQVbz8MfPznHM6Z3wkZFVLTPpSFxaXldXCWnF9Y3NrW93ZbYkg4phYOGAB7zhIEZ9YkqGemEnCDPYaTtjC+m9fYt4YIGflNOQmJ7aOhTl2IkU6uvHvQ8JEeOG98kZ3O0kuvmnKOkr5a0spYJ/gU9hxLI1eir71BgCOP+BIzJERX10Jpx4hLihlJir1IkBDhMRqSbo+8oiw4+wrCTxKnQF0A54+X8LM/T4RI0+IiekndMLxe/a1Pyv1o2ka9ox9cNIEh/PFrkRgzKA01zgHKCJZukgDCn6a0QjxBHWKbpFbMQaqZumFWYQfXEnIFRqX2F0DLKeqVsXFVL9fM8jgLYB4fgGOjgFNTBJWgAC2BwBx7AE3hW7pVH5UV5nbUuKPnMHvgh5e0TvBib1g=</latexit> <latexit sha1_base64="6amS2/naPupSAaYrU9IA1xTets=">ACHicbZDLSsNAFIYnXmu9RV26cLAIrkqSFk0XQtGNywqmLbShTKaTdujk4sxEKCFLN76KGxeKuPUR3Pk2Jmkq3n4Y+PjPOZwzvxMyKqSmfSgLi0vLK6ultfL6xubWtrqz2xZBxDGxcMAC3nWQIz6xJUMtINOUGew0jHmVxk9c4t4YIG/rWchsT20MinLsVIptZAPeh7SI4dN46SszlayZxukoFa0apaLvgX9AIqoFBroL73hwGOPOJLzJAQPV0LpR0jLilmJCn3I0FChCdoRHop+sgjwo7zjyTwKHWG0A14+nwJc/f7RIw8Iaek3ZmF4rftcz8r9aLpGvaMfXDSBIfzxa5EYMygFkqcEg5wZJNU0CY0/RWiMeIyzT7Mp5CA1TN8w6zKF+Ys7AqDW+QmgbVb1WNa7qleZ5EUcJ7INDcAx0cAqa4BK0gAUwuAMP4Ak8K/fKo/KivM5aF5RiZg/8kPL2CUulmxA=</latexit> <latexit sha1_base64="MVPDk0jXGWEVDRod+RSwgkbUkfg=">ACnicbZC7TsMwFIYdrqXcCowshgqJqUraqrRbBQtjQfQiNaFyXKe16jBdpCqKDMLr8LCAEKsPAEb0NuQtx+ydLn/5wjH/+2z6hUuv6hLSwuLa+sFtaK6xubW9ulnd2e9AKBSRd7zBMDG0nCKCdRUjA18Q5NqM9O3ZWVLv3xIhqcev1NwnlosmnDoUIxVbo9KB6SI1tZ3wJjIpzy52eBldh6akLmzo0ahU1it6KvgXjBzKIFdnVHo3x4OXMIVZkjKoaH7ygqRUBQzEhXNQBIf4RmakGMHLlEWmH6lQgexc4YOp6ID1cwdb9PhMiVcu7acWeyq/xdS8z/asNAOU0rpNwPFOE4e8gJGFQeTHKBYyoIVmweA8KCxrtCPEUCYRWnV0xDaDWNarMOU6g3mhlUa62vEHrVilGrVC/q5fZpHkcB7INDcAwMcALa4Bx0QBdgcAcewBN41u61R+1Fe81aF7R8Zg/8kPb2CQGPm1k=</latexit> <latexit sha1_base64="ZFMGVflohNOk+NWCQZ+vUJdZDU=">ACXicbZC7TsMwFIYdrqXcAowsFhUSA6qcNoJ2q2BhLIhepDZUju0Vh0nsh2kKsrKwquwMIAQK2/AxtuQphHi9kuWPv/nHPn4d0POlEbow1hYXFpeWS2sFdc3Nre2zZ3dtgoiSWiLBDyQXRcrypmgLc0p91QUuy7nHbcyfms3rmlUrFAXOtpSB0fjwTzGME6tQYm7PtYj10vjpI+E/OLG18lN7GNjhFCycAsoTLKBP+ClUMJ5GoOzPf+MCRT4UmHCvVs1ConRhLzQinSbEfKRpiMsEj2ktRYJ8qJ85+ksD1BlCL5DpERpm7veJGPtKTX037Zytqn7XZuZ/tV6kvZoTMxFGmgoyf8iLONQBnMUCh0xSovk0BUwkS3eFZIwlJjoNr5iFUK9ZlZoNM7BPanOoVOtfIbQrZatarlzapcZHkcB7IMDcAQscAoa4AI0QsQcAcewBN4Nu6NR+PFeJ23Lhj5zB74IePtEz2emkg=</latexit> Existing Exi ng Work: k: Mode del Reduc duction High u ∈ R 40 , 000 Dimensional System q = U T u u = Uq Low q ∈ R ∼ 60 Dimensional System Reduced Coordinates

  10. <latexit sha1_base64="IvQEuEw+s24Pj1dZgj4dp1VEpY=">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</latexit> <latexit sha1_base64="drzuGJFNSle1VIVpOsE9BZr27SA=">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</latexit> Mod Model R Reduction on Replace high-dimensional problem with low-dimensional u n +1 = argmin E ( u ) u Big Becomes q n +1 = argmin E ( Uq ) q Small

  11. St Static Sol c Solve E Examp mple

  12. <latexit sha1_base64="IvQEuEw+s24Pj1dZgj4dp1VEpY=">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</latexit> Where does come from? ( Uq

  13. Mod Model R Reduction on - Ex Exampl ple

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