27 research outputs found
Deep Conservation: A latent-dynamics model for exact satisfaction of physical conservation laws
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
FAST AND OPTIMAL SOLUTION ALGORITHMS FOR PARAMETERIZED PARTIAL DIFFERENTIAL EQUATIONS
This dissertation presents efficient and optimal numerical algorithms for the solution of parameterized partial differential equations (PDEs) in the context of stochastic Galerkin discretization. The stochastic Galerkin method often leads to a large coupled system of algebraic equations, whose solution is computationally expensive to compute using traditional solvers. For efficient computation of such solutions, we present low-rank iterative solvers, which compute low-rank approximations to the solutions of those systems while not losing much accuracy. We first introduce a low-rank iterative solver for linear systems obtained from the stochastic Galerkin discretization of linear elliptic parameterized PDEs. Then we present a low-rank nonlinear iterative solver for efficiently computing approximate solutions of nonlinear parameterized PDEs, the incompressible Navier–Stokes equations.
Along with the computational issue, the stochastic Galerkin method suffers from an optimality issue. The method, in general, does not minimize the solution error in any measure. To address this issue, we present an optimal projection method, a least-squares Petrov–Galerkin (LSPG) method. The proposed method is optimal in the sense that it produces the solution that minimizes a weighted l2-norm of the solution error over all solutions in a given finite-dimensional subspace. The method can be adapted to minimize the solution error in different weighted l2-norms by simply choosing a specific weighting function within the least-squares formulation
Reduced-order modeling for parameterized PDEs via implicit neural representations
We present a new data-driven reduced-order modeling approach to efficiently
solve parametrized partial differential equations (PDEs) for many-query
problems. This work is inspired by the concept of implicit neural
representation (INR), which models physics signals in a continuous manner and
independent of spatial/temporal discretization. The proposed framework encodes
PDE and utilizes a parametrized neural ODE (PNODE) to learn latent dynamics
characterized by multiple PDE parameters. PNODE can be inferred by a
hypernetwork to reduce the potential difficulties in learning PNODE due to a
complex multilayer perceptron (MLP). The framework uses an INR to decode the
latent dynamics and reconstruct accurate PDE solutions. Further, a
physics-informed loss is also introduced to correct the prediction of unseen
parameter instances. Incorporating the physics-informed loss also enables the
model to be fine-tuned in an unsupervised manner on unseen PDE parameters. A
numerical experiment is performed on a two-dimensional Burgers equation with a
large variation of PDE parameters. We evaluate the proposed method at a large
Reynolds number and obtain up to speedup of O(10^3) and ~1% relative error to
the ground truth values.Comment: 9 pages, 5 figures, Machine Learning and the Physical Sciences
Workshop, NeurIPS 202
DPM: A Novel Training Method for Physics-Informed Neural Networks in Extrapolation
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) [Raissi et al., J. Comput. Phys.,
378:686--707, 2019] 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.Comment: Accepted by AAAI 202