{"title": "Fast Neural Network Emulation of Dynamical Systems for Computer Animation", "book": "Advances in Neural Information Processing Systems", "page_first": 882, "page_last": 888, "abstract": null, "full_text": "Fast Neural Network Emulation of Dynamical \n\nSystems for Computer Animation \n\nRadek Grzeszczuk 1 \n\nDemetri Terzopoulos 2 \n\nGeoffrey Hinton 2 \n\n1 Intel Corporation \n\nMicrocomputer Research Lab \n2200 Mission College Blvd. \nSanta Clara, CA 95052, USA \n\n2 University of Toronto \n\nDepartment of Computer Science \n\n10 King's College Road \n\nToronto, ON M5S 3H5, Canada \n\nAbstract \n\nComputer animation through the numerical simulation of physics-based \ngraphics models offers unsurpassed realism, but it can be computation(cid:173)\nally demanding. This paper demonstrates the possibility of replacing the \nnumerical simulation of nontrivial dynamic models with a dramatically \nmore efficient \"NeuroAnimator\" that exploits neural networks. Neu(cid:173)\nroAnimators are automatically trained off-line to emulate physical dy(cid:173)\nnamics through the observation of physics-based models in action. De(cid:173)\npending on the model, its neural network emulator can yield physically \nrealistic animation one or two orders of magnitude faster than conven(cid:173)\ntional numerical simulation. We demonstrate NeuroAnimators for a va(cid:173)\nriety of physics-based models. \n\n1 \n\nIntroduction \n\nAnimation based on physical principles has been an influential trend in computer graphics \nfor over a decade (see, e.g., [1, 2, 3]). This is not only due to the unsurpassed realism \nthat physics-based techniques offer. In conjunction with suitable control and constraint \nmechanisms, physical models also facilitate the production of copious quantities of real(cid:173)\nistic animation in a highly automated fashion. Physics-based animation techniques are \nbeginning to find their way into high-end commercial systems. However, a well-known \ndrawback has retarded their broader penetration--compared to geometric models, physical \nmodels typically entail formidable numerical simulation costs. \n\nThis paper proposes a new approach to creating physically realistic animation that differs \n\n\fEmulation for Animation \n\n883 \n\nradically from the conventional approach of numerically simulating the equations of mo(cid:173)\ntion of physics-based models. We replace physics-based models by fast emulators which \nautomatically learn to produce similar motions by observing the models in action. Our \nemulators have a neural network structure, hence we dub them NeuroAnimators. \n\nOur work is inspired in part by that of Nguyen and Widrow [4]. Their \"truck backer-upper\" \ndemonstrated the neural network based approximation and control of a nonlinear kinematic \nsystem. We introduce several generalizations that enable us to tackle a variety of complex, \nfully dynamic models in the context of computer animation. Connectionist approximations \nof dynamical systems have been also been applied to robot control (see, e.g., [5,6]). \n\n2 The NeuroAnimator Approach \n\nOur approach is motivated by the following considerations: Whether we are dealing with \nrigid [2], articulated [3], or nonrigid [I] dynamic animation models, the numerical sim(cid:173)\nulation of the associated equations of motion leads to the computation of a discrete-time \ndynamical system of the form StHt = ~[St, Ut, ft ]. These (generally nonlinear) equations \nexpress the vector St+8t of state variables of the system (values of the system's degrees of \nfreedom and their velocities) at time t + r5t in the future as a function ~ of the state vector \nSt, the vector Ut of control inputs, and the vector ft of external forces acting on the system \nat time t. \n\nPhysics-based animation through the numerical simulation of a dynamical system requires \nthe evaluation of the map ~ at every timestep, which usually involves a non-trivial compu(cid:173)\ntation. Evaluating ~ using explicit time integration methods incurs a computational cost of \nO(N) operations, where N is proportional to the dimensionality of the state space. Unfor(cid:173)\ntunately, for many dynamic models of interest, explicit methods are plagued by instability, \nnecessitating numerous tiny timesteps r5t per unit simulation time. Alternatively, implicit \ntime-integration methods usually permit larger timesteps, but they compute ~ by solving a \nsystem of N algebraic equations, generally incurring a cost of O( N 3 ) per timestep. \n\nIs it possible to replace the conventional numerical simulator by a significantly cheaper \nalternative? A crucial realization is that the substitute, or emulator, need not compute \nthe map ~ exactly, but merely approximate it to a degree of precision that preserves the \nperceived faithfulness of the resulting animation to the simulated dynamics of the physical \nmodel. Neural networks offer a general mechanism for approximating complex maps in \nhigher dimensional spaces [7].1 Our premise is that, to a sufficient degree of accuracy and \nat significant computational savings, trained neural networks can approximate maps ~ not \njust for simple dynamical systems, but also for those associated with dynamic models that \nare among the most complex reported in the graphics literature to date. \n\nThe NeuroAnimator, which uses neural networks to emulate physics-based animation, \nlearns an approximation to the dynamic model by observing instances of state transitions, \nas well as control inputs and/or external forces that cause these transitions. By generalizing \nfrom the sparse examples presented to it, a trained NeuroAnimator can emulate an infinite \nvariety of continuous animations that it has never actually seen. Each emulation step costs \nonly O(N2) operations, but it is possible to gain additional efficiency relative to a numer(cid:173)\nical simulator by training neural networks to approximate a lengthy chain of evaluations \nof the discrete-time dynamical system. Thus, the emulator network can perform \"super \nI Note that q, is in general a high-dimensional map from RS+u+ f t---7 RS, where s, u, and f denote \n\nthe dimensionalities of the state, control, and external force vectors. \n\n\f884 \n\nR. Grzeszczuk, D. Terzopoulos and G. E. Hinton \n\ntimesteps\" b.t = n6t, typically one or two orders of magnitude larger than 6t for the com(cid:173)\npeting implicit time-integration scheme, thereby achieving outstanding efficiency without \nserious loss of accuracy. \n\n3 From Physics-Based Models to NeuroAnimators \n\nOur task is to construct neural networks that approximate
, with a single layer \nof sigmoidal hidden units, to predict future states using super time steps b.t = n6t while \ncontaining the approximation error so as not to appreciably degrade the physical realism of \nthe resulting animation. The basic emulation step is St+~t = N