Physics constrained learning for data-driven inverse modeling from sparse observations

Xu, Kailai; Darve, Eric

doi:10.1016/j.jcp.2021.110938

Cited by 32 publications

(25 citation statements)

References 47 publications

(30 reference statements)

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“…It is thus clear that, given a single solution u(x, t) corresponding to an initial data u(0, t), the underlying constant coefficient PDE is identifiable, i.e., there exists a unique set of parameters p α such that ∂ t u = −Lu if and only if (47) and ( 48) admit unique solutions for c α , which are coefficients of two polynomials. If t 2 − t 1 > 0 is small enough, the phase ambiguity in (48) is removed. Hence one can apply standard polynomial regression result to this problem in spectral domain.…”

Section: Pde Identification With Constant Coefficientmentioning

confidence: 99%

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How much can one learn a partial differential equation from its solution?

He¹,

Zhao²,

Zhong³

2022

Preprint

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In this work we study the problem about learning a partial differential equation (PDE) from its solution data. PDEs of various types are used as examples to illustrate how much the solution data can reveal the PDE operator depending on the underlying operator and initial data. A data driven and data adaptive approach based on local regression and global consistency is proposed for stable PDE identification. Numerical experiments are provided to verify our analysis and demonstrate the performance of the proposed algorithms.

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Section: Pde Identification With Constant Coefficientmentioning

confidence: 99%

“…Many methods have been proposed for PDE learning [6,10,16,18,21,22,26,27,30,33,37,38,41,42,43,44,46,47,48]. In general, there are two approaches.…”

Section: Introductionmentioning

confidence: 99%

How much can one learn a partial differential equation from its solution?

He¹,

Zhao²,

Zhong³

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…37,38,39,40,41 Physics-constrained machine learning, on the other hand, enforces the physical relations more strictly by propagating numerical gradients through numerical PDE schemes using automatic differentiation. 9,10,42,11,43 In addition, neural networks as highly flexible functions are used as surrogate models for implicit physical relations to provide end-to-end approximate simulation outcomes 44,7,45 or as augmentations and corrections to traditional schemes. 8…”

Section: Scientific Machine Learningmentioning

confidence: 99%

“…3,4,5 Data-driven learning has also demonstrated great potential to facilitate efficient engineering simulations. 6,7,8,9,10,11 However, many tools are designed for specific tasks and cannot be easily generalized to other engineering systems. 12 Moreover, although uncertainty quantification is required in many applications, it may not be naturally produced by the approximation scheme itself.…”

Section: Introductionmentioning

confidence: 99%

Probabilistic partition of unity networks for high-dimensional regression problems

Fan¹,

Trask²,

D’Elia³

et al. 2022

Preprint

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We explore the probabilistic partition of unity network (PPOU-Net) model 1,2 in the context of high-dimensional regression problems. With the PPOU-Nets, the target function for any given input is approximated by a mixture of experts model, where each cluster is associated with a fixed-degree polynomial. The weights of the clusters are determined by a DNN that defines a partition of unity. The weighted average of the polynomials approximates the target function and produces uncertainty quantification naturally. Our training strategy leverages automatic differentiation and the expectation maximization (EM) algorithm. During the training, we (i) apply gradient descent to update the DNN coefficients; (ii) update the polynomial coefficients using weighted least-squares solves; and (iii) compute the variance of each cluster according to a closed-form formula derived from the EM algorithm. The PPOU-Nets consistently outperform the baseline fully-connected neural networks of comparable sizes in numerical experiments of various data dimensions. We also explore the proposed model in applications of quantum computing, where the PPOU-Nets act as surrogate models for cost landscapes associated with variational quantum circuits.

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“…Very recently, the use of DNNs for solving PDEs inverse problems also started to receive attention, and existing methods can roughly be divided into two groups: supervised [62,37,27] and unsupervised [6,7,57,72]. The methods in the former group relies on the availability of paired training data, and are essentially concerned with learning the forward operators or its (regularized) inverses, and the methods in the latter group exploit essentially the extraordinary expressivity as universal function approximators.…”

Section: Introductionmentioning

confidence: 99%

Imaging conductivity from current density magnitude using neural networks*

Jin

2022

Inverse Problems

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Conductivity imaging represents one of the most important tasks in medical imaging. In this work we develop a neural network based reconstruction technique for imaging the conductivity from the magnitude of the internal current density. It is achieved by formulating the problem as a relaxed weighted least-gradient problem, and then approximating its minimizer by standard fully connected feedforward neural networks. We derive bounds on two components of the generalization error, i.e., approximation error and statistical error, explicitly in terms of properties of the neural networks (e.g., depth, total number of parameters, and the bound of the network parameters). We illustrate the performance and distinct features of the approach on several numerical experiments. Numerically, it is observed that the approach enjoys remarkable robustness with respect to the presence of data noise.

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Physics constrained learning for data-driven inverse modeling from sparse observations

Cited by 32 publications

References 47 publications

How much can one learn a partial differential equation from its solution?

How much can one learn a partial differential equation from its solution?

Probabilistic partition of unity networks for high-dimensional regression problems

Imaging conductivity from current density magnitude using neural networks*

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