## Deep Conservation: A Latent-Dynamics Model for Exact Satisfaction of Physical Conservation Laws [Slides]

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35th AAAI Conference on Artificial Intelligence, AAAI 2021

We present a method for learning dynamics of complex physical processes described by time-dependent nonlinear partial differential equations (PDEs). Our particular interest lies in extrapolating solutions in time beyond the range of temporal domain used in training. Our choice for a baseline method is physics-informed neural network (PINN) because the method parameterizes not only the solutions, but also the equations that describe the dynamics of physical processes. We demonstrate that PINN performs poorly on extrapolation tasks in many benchmark problems. To address this, we propose a novel method for better training PINN and demonstrate that our newly enhanced PINNs can accurately extrapolate solutions in time. Our method shows up to 72% smaller errors than existing methods in terms of the standard L2-norm metric.

35th AAAI Conference on Artificial Intelligence, AAAI 2021

This work proposes an approach for latent-dynamics learning that exactly enforces physical conservation laws. The method comprises two steps. First, the method computes a low-dimensional embedding of the high-dimensional dynamical-system state using deep convolutional autoencoders. This defines a low-dimensional nonlinear manifold on which the state is subsequently enforced to evolve. Second, the method defines a latent-dynamics model that associates with the solution to a constrained optimization problem. Here, the objective function is defined as the sum of squares of conservation-law violations over control volumes within a finite-volume discretization of the problem; nonlinear equality constraints explicitly enforce conservation over prescribed subdomains of the problem. Under modest conditions, the resulting dynamics model guarantees that the time-evolution of the latent state exactly satisfies conservation laws over the prescribed subdomains.

Proceedings - 2019 IEEE International Conference on Big Data, Big Data 2019

Knowledge graph embedding (KGE) learns latent vector representations of named entities (i.e., vertices) and relations (i.e., edge labels) of knowledge graphs. Herein, we address two problems in KGE. First, relations may belong to one or multiple categories, such as functional, symmetric, transitive, reflexive, and so forth; thus, relation categories are not exclusive. Some relation categories cause non-trivial challenges for KGE. Second, we found that zero gradients happen frequently in many translation based embedding methods such as TransE and its variations. To solve these problems, we propose i) converting a knowledge graph into a bipartite graph, although we do not physically convert the graph but rather use an equivalent trick; ii) using multiple vector representations for a relation; and iii) using a new hinge loss based on energy ratio(rather than energy gap) that does not cause zero gradients. We show that our method significantly improves the quality of embedding.

This work proposes an approach for latent dynamics learning that exactly enforces physical conservation laws. The method comprises two steps. First, we compute a low-dimensional embedding of the high-dimensional dynamical-system state using deep convolutional autoencoders. This defines a low-dimensional nonlinear manifold on which the state is subsequently enforced to evolve. Second, we define a latent dynamics model that associates with a constrained optimization problem. Specifically, the objective function is defined as the sum of squares of conservation-law violations over control volumes in a finite-volume discretization of the problem; nonlinear equality constraints explicitly enforce conservation over prescribed subdomains of the problem. The resulting dynamics modelâ€”which can be considered as a projection-based reduced-order modelâ€”ensures that the time-evolution of the latent state exactly satisfies conservation laws over the prescribed subdomains. In contrast to existing methods for latent dynamics learning, this is the only method that both employs a nonlinear embedding and computes dynamics for the latent state that guarantee the satisfaction of prescribed physical properties. Numerical experiments on a benchmark advection problem illustrate the method's ability to significantly reduce the dimensionality while enforcing physical conservation.

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SIAM/ASA Journal on Uncertainty Quantification

We study two inexact methods for solutions of random eigenvalue problems in the context of spectral stochastic finite elements. In particular, given a parameter-dependent, symmetric matrix operator, the methods solve for eigenvalues and eigenvectors represented using polynomial chaos expansions. Both methods are based on the stochastic Galerkin formulation of the eigenvalue problem and they exploit its Kronecker-product structure. The first method is an inexact variant of the stochastic inverse subspace iteration [B. Sousedfk, H. C. Elman, SIAM/ASA Journal on Uncertainty Quantification 4(1), pp. 163-189, 2016]. The second method is based on an inexact variant of Newton iteration. In both cases, the problems are formulated so that the associated stochastic Galerkin matrices are symmetric, and the corresponding linear problems are solved using preconditioned Krylov subspace methods with several novel hierarchical preconditioners. The accuracy of the methods is compared with that of Monte Carlo and stochastic collocation, and the effectiveness of the methods is illustrated by numerical experiments.

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