We explore using neural operators, or neural network representations of nonlinear maps between function spaces, to accelerate infinite-dimensional Bayesian inverse problems (BIPs) with models governed by nonlinear parametric partial differential equations (PDEs). Neural operators have gained significant attention in recent years for their ability to approximate the parameter-to-solution maps defined by PDEs using as training data solutions of PDEs at a limited number of parameter samples. The computational cost of BIPs can be drastically reduced if the large number of PDE solves required for posterior characterization are replaced with evaluations of trained neural operators. However, reducing error in the resulting BIP solutions via reducing the approximation error of the neural operators in training can be challenging and unreliable. We provide an a priori error bound result that implies certain BIPs can be ill-conditioned to the approximation error of neural operators, thus leading to inaccessible accuracy requirements in training. To reliably deploy neural operators in BIPs, we consider a strategy for enhancing the performance of neural operators, which is to correct the prediction of a trained neural operator by solving a linear variational problem based on the PDE residual. We show that a trained neural operator with error correction can achieve a quadratic reduction of its approximation error, all while retaining substantial computational speedups of posterior sampling when models are governed by highly nonlinear PDEs. The strategy is applied to two numerical examples of BIPs based on a nonlinear reaction--diffusion problem and deformation of hyperelastic materials. We demonstrate that posterior representations of the two BIPs produced using trained neural operators are greatly and consistently enhanced by error correction.
相關內容
We explore a method of statistical estimation called Maximum Entropy on the Mean (MEM) which is based on an information-driven criterion that quantifies the compliance of a given point with a reference prior probability measure. At the core of this approach lies the MEM function which is a partial minimization of the Kullback-Leibler divergence over a linear constraint. In many cases, it is known that this function admits a simpler representation (known as the Cram\'er rate function). Via the connection to exponential families of probability distributions, we study general conditions under which this representation holds. We then address how the associated MEM estimator gives rise to a wide class of MEM-based regularized linear models for solving inverse problems. Finally, we propose an algorithmic framework to solve these problems efficiently based on the Bregman proximal gradient method, alongside proximal operators for commonly used reference distributions. The article is complemented by a software package for experimentation and exploration of the MEM approach in applications.
In this paper, we consider the optimization problem \scpl (\scp), which is to find a minimum cost subset of a ground set $U$ such that the value of a submodular function $f$ is above a threshold $\tau$. In contrast to most existing work on \scp, it is not assumed that $f$ is monotone. Two bicriteria approximation algorithms are presented for \scp that, for input parameter $0 < \epsilon < 1$, give $O( 1 / \epsilon^2 )$ ratio to the optimal cost and ensures the function $f$ is at least $\tau(1 - \epsilon)/2$. A lower bound shows that under the value query model shows that no polynomial-time algorithm can ensure that $f$ is larger than $\tau/2$. Further, the algorithms presented are scalable to large data sets, processing the ground set in a stream. Similar algorithms developed for \scp also work for the related optimization problem of \smpl (\smp). Finally, the algorithms are demonstrated to be effective in experiments involving graph cut and data summarization functions.
The goal of model compression is to reduce the size of a large neural network while retaining a comparable performance. As a result, computation and memory costs in resource-limited applications may be significantly reduced by dropping redundant weights, neurons, or layers. There have been many model compression algorithms proposed that provide impressive empirical success. However, a theoretical understanding of model compression is still limited. One problem is understanding if a network is more compressible than another of the same structure. Another problem is quantifying how much one can prune a network with theoretically guaranteed accuracy degradation. In this work, we propose to use the sparsity-sensitive $\ell_q$-norm ($0<q<1$) to characterize compressibility and provide a relationship between soft sparsity of the weights in the network and the degree of compression with a controlled accuracy degradation bound. We also develop adaptive algorithms for pruning each neuron in the network informed by our theory. Numerical studies demonstrate the promising performance of the proposed methods compared with standard pruning algorithms.
This paper presents CQT-Diff, a data-driven generative audio model that can, once trained, be used for solving various different audio inverse problems in a problem-agnostic setting. CQT-Diff is a neural diffusion model with an architecture that is carefully constructed to exploit pitch-equivariant symmetries in music. This is achieved by preconditioning the model with an invertible Constant-Q Transform (CQT), whose logarithmically-spaced frequency axis represents pitch equivariance as translation equivariance. The proposed method is evaluated with objective and subjective metrics in three different and varied tasks: audio bandwidth extension, inpainting, and declipping. The results show that CQT-Diff outperforms the compared baselines and ablations in audio bandwidth extension and, without retraining, delivers competitive performance against modern baselines in audio inpainting and declipping. This work represents the first diffusion-based general framework for solving inverse problems in audio processing.
Vector autoregressions (VARs) are popular in analyzing economic time series. However, VARs can be over-parameterized if the numbers of variables and lags are moderately large. Tensor VAR, a recent solution to overparameterization, treats the coefficient matrix as a third-order tensor and estimates the corresponding tensor decomposition to achieve parsimony. In this paper, the inference of Tensor VARs is inspired by the literature on factor models. Firstly, we determine the rank by imposing the Multiplicative Gamma Prior to margins, i.e. elements in the decomposition, and accelerate the computation with an adaptive inferential scheme. Secondly, to obtain interpretable margins, we propose an interweaving algorithm to improve the mixing of margins and introduce a post-processing procedure to solve column permutations and sign-switching issues. In the application of the US macroeconomic data, our models outperform standard VARs in point and density forecasting and yield interpretable results consistent with the US economic history.
We correct two errors in our paper [4]. First error concerns the definition of the SVI solution, where a boundary term which arises due to the Dirichlet boundary condition, was not included. The second error concerns the discrete estimate [4, Lemma 4.4], which involves the discrete Laplace operator. We provide an alternative proof of the estimate in spatial dimension $d=1$ by using a mass lumped version of the discrete Laplacian. Hence, after a minor modification of the fully discrete numerical scheme the convergence in $d=1$ follows along the lines of the original proof. The convergence proof of the time semi-discrete scheme, which relies on the continuous counterpart of the estimate [4, Lemma 4.4], remains valid in higher spatial dimension. The convergence of the fully discrete finite element scheme from [4] in any spatial dimension is shown in [3] by using a different approach.
In this paper I will present a novel way of combining proof net proof search with neural networks. It contrasts with the 'standard' approach which has been applied to proof search in type-logical grammars in various different forms. In the standard approach, we first transform words to formulas (supertagging) then match atomic formulas to obtain a proof. I will introduce an alternative way to split the task into two: first, we generate the graph structure in a way which guarantees it corresponds to a lambda-term, then we obtain the detailed structure using vertex labelling. Vertex labelling is a well-studied task in graph neural networks, and different ways of implementing graph generation using neural networks will be explored.
We consider optimal experimental design (OED) for Bayesian nonlinear inverse problems governed by partial differential equations (PDEs) under model uncertainty. Specifically, we consider inverse problems in which, in addition to the inversion parameters, the governing PDEs include secondary uncertain parameters. We focus on problems with infinite-dimensional inversion and secondary parameters and present a scalable computational framework for optimal design of such problems. The proposed approach enables Bayesian inversion and OED under uncertainty within a unfied framework. We build on the Bayesian approximation error (BAE) framework, to incorporate modeling uncertainties in the Bayesian inverse problem, and methods for A-optimal design of infinite-dimensional Bayesian nonlinear inverse problems. Specifically, a Gaussian approximation to the posterior at the maximum a posteriori probability point is used to define an uncertainty aware OED objective that is tractable to evaluate and optimize. In particular, the OED objective can be computed at a cost, in the number of PDE solves, that does not grow with the dimension of the discretized inversion and secondary parameters. The OED problem is formulated as a binary bilevel PDE constrained optimization problem and a greedy algorithm, which provides a pragmatic approach, is used to find optimal designs. We demonstrate the effectiveness of the proposed approach for a model inverse problem governed by an elliptic PDE on a three-dimensional domain. Our computational results also highlight the pitfalls of ignoring modeling uncertainties in the OED and/or inference stages.
When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.
Deep convolutional neural networks (CNNs) have recently achieved great success in many visual recognition tasks. However, existing deep neural network models are computationally expensive and memory intensive, hindering their deployment in devices with low memory resources or in applications with strict latency requirements. Therefore, a natural thought is to perform model compression and acceleration in deep networks without significantly decreasing the model performance. During the past few years, tremendous progress has been made in this area. In this paper, we survey the recent advanced techniques for compacting and accelerating CNNs model developed. These techniques are roughly categorized into four schemes: parameter pruning and sharing, low-rank factorization, transferred/compact convolutional filters, and knowledge distillation. Methods of parameter pruning and sharing will be described at the beginning, after that the other techniques will be introduced. For each scheme, we provide insightful analysis regarding the performance, related applications, advantages, and drawbacks etc. Then we will go through a few very recent additional successful methods, for example, dynamic capacity networks and stochastic depths networks. After that, we survey the evaluation matrix, the main datasets used for evaluating the model performance and recent benchmarking efforts. Finally, we conclude this paper, discuss remaining challenges and possible directions on this topic.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191