20 research outputs found
Global convergence of optimized adaptive importance samplers
We analyze the optimized adaptive importance sampler (OAIS) for performing
Monte Carlo integration with general proposals. We leverage a classical result
which shows that the bias and the mean-squared error (MSE) of the importance
sampling scales with the -divergence between the target and the
proposal and develop a scheme which performs global optimization of
-divergence. While it is known that this quantity is convex for
exponential family proposals, the case of the general proposals has been an
open problem. We close this gap by utilizing the nonasymptotic bounds for
stochastic gradient Langevin dynamics (SGLD) for the global optimization of
-divergence and derive nonasymptotic bounds for the MSE by leveraging
recent results from non-convex optimization literature. The resulting AIS
schemes have explicit theoretical guarantees that are uniform-in-time.Comment: Accepted to Foundations of Data Science (FoDS), 2024, to appea
Adaptively Optimised Adaptive Importance Samplers
We introduce a new class of adaptive importance samplers leveraging adaptive
optimisation tools, which we term AdaOAIS. We build on Optimised Adaptive
Importance Samplers (OAIS), a class of techniques that adapt proposals to
improve the mean-squared error of the importance sampling estimators by
parameterising the proposal and optimising the -divergence between the
target and the proposal. We show that a naive implementation of OAIS using
stochastic gradient descent may lead to unstable estimators despite its
convergence guarantees. To remedy this shortcoming, we instead propose to use
adaptive optimisers (such as AdaGrad and Adam) to improve the stability of the
OAIS. We provide convergence results for AdaOAIS in a similar manner to OAIS.
We also provide empirical demonstration on a variety of examples and show that
AdaOAIS lead to stable importance sampling estimators in practice.Comment: This work has been submitted to the IEEE for possible publication.
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Fully probabilistic deep models for forward and inverse problems in parametric PDEs
We introduce a physics-driven deep latent variable model (PDDLVM) to learn
simultaneously parameter-to-solution (forward) and solution-to-parameter
(inverse) maps of parametric partial differential equations (PDEs). Our
formulation leverages conventional PDE discretization techniques, deep neural
networks, probabilistic modelling, and variational inference to assemble a
fully probabilistic coherent framework. In the posited probabilistic model,
both the forward and inverse maps are approximated as Gaussian distributions
with a mean and covariance parameterized by deep neural networks. The PDE
residual is assumed to be an observed random vector of value zero, hence we
model it as a random vector with a zero mean and a user-prescribed covariance.
The model is trained by maximizing the probability, that is the evidence or
marginal likelihood, of observing a residual of zero by maximizing the evidence
lower bound (ELBO). Consequently, the proposed methodology does not require any
independent PDE solves and is physics-informed at training time, allowing the
real-time solution of PDE forward and inverse problems after training. The
proposed framework can be easily extended to seamlessly integrate observed data
to solve inverse problems and to build generative models. We demonstrate the
efficiency and robustness of our method on finite element discretized
parametric PDE problems such as linear and nonlinear Poisson problems, elastic
shells with complex 3D geometries, and time-dependent nonlinear and
inhomogeneous PDEs using a physics-informed neural network (PINN)
discretization. We achieve up to three orders of magnitude speed-up after
training compared to traditional finite element method (FEM), while outputting
coherent uncertainty estimates
Random Grid Neural Processes for Parametric Partial Differential Equations
We introduce a new class of spatially stochastic physics and data informed
deep latent models for parametric partial differential equations (PDEs) which
operate through scalable variational neural processes. We achieve this by
assigning probability measures to the spatial domain, which allows us to treat
collocation grids probabilistically as random variables to be marginalised out.
Adapting this spatial statistics view, we solve forward and inverse problems
for parametric PDEs in a way that leads to the construction of Gaussian process
models of solution fields. The implementation of these random grids poses a
unique set of challenges for inverse physics informed deep learning frameworks
and we propose a new architecture called Grid Invariant Convolutional Networks
(GICNets) to overcome these challenges. We further show how to incorporate
noisy data in a principled manner into our physics informed model to improve
predictions for problems where data may be available but whose measurement
location does not coincide with any fixed mesh or grid. The proposed method is
tested on a nonlinear Poisson problem, Burgers equation, and Navier-Stokes
equations, and we provide extensive numerical comparisons. We demonstrate
significant computational advantages over current physics informed neural
learning methods for parametric PDEs while improving the predictive
capabilities and flexibility of these models