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
Wh Why y de deformabl ble si simul ulation? n? [Ziva Dynamics]
Re Research Question Can we use machine learning to accelerate hyperelastic simulation?
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
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
Ou Our Approach Build on the vast literature of Model Reduction Simulate in nonlinear latent space using the true equations of motion
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<latexit sha1_base64="drzuGJFNSle1VIVpOsE9BZr27SA=">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</latexit> <latexit sha1_base64="S7wdZrM4q3SeSHGz4kwVKdgAXQ=">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</latexit> <latexit sha1_base64="drzuGJFNSle1VIVpOsE9BZr27SA=">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</latexit> Solver Sol Fast and stable solution: Implicit Euler as a minimization problem Inertia Term New configuration V ( u ) + I ( u , u n , ˙ u n +1 = argmin u n ) u Elastic Potential Previous State u n +1 = argmin in E ( u ) E ( u ) Objective Function u Solve using pre-conditioned quasi-newton solver like L-BFGS
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<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
St Static Sol c Solve E Examp mple
<latexit sha1_base64="IvQEuEw+s24Pj1dZgj4dp1VEpY=">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</latexit> Where does come from? ( Uq
Mod Model R Reduction on - Ex Exampl ple
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