Fast PDE-constrained optimization via self-supervised operator learning

Wang, Sifan; Bhouri, Mohamed Aziz; Perdikaris, Paris

doi:10.48550/arxiv.2110.13297

Cited by 5 publications

(6 citation statements)

References 43 publications

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“…The trunk network is composed of 5 hidden layers with 300 neurons in each layer and 150 neurons in the output layer. The loss function for known data is similar to (25). Taken into account the physics informed MOONet and α = 1, the corresponding loss function is expressed as follows…”

Section: P R E P R I N T N O T P E E R R E V I E W E Dmentioning

confidence: 99%

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Deep Multi-Input and Multi-Output Operator Networks Method for Optimal Control of Pdes

Yong

Luo

Sun

2023

Preprint

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Section: P R E P R I N T N O T P E E R R E V I E W E Dmentioning

confidence: 99%

“…Barry-Straume et al use a two-stage framework to solve PDE-constrained optimization problems [24]. Wang et al use physics-informed deep operator networks (DeepONets) framework to learn the solution operator of parametric PDEs, which builds a surrogate for solving PDE-constrained optimization problems [25].…”

Section: Introductionmentioning

confidence: 99%

Deep Multi-Input and Multi-Output Operator Networks Method for Optimal Control of Pdes

Yong

Luo

Sun

2023

Preprint

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“…We note that the same neural operator can also be deployed to accelerate Bayesian inverse problems governed by the same model but defined by possibly a variety of different noise models, observation operators, or data, which leads to additional computational cost reduction via the amortization of the model deployment in many different problems. Neural operators have also observed success in their deployment as surrogates for accelerating so-called "outer-loop" problems, such as inverse problems [36], Bayesian optimal experimental design [72], PDE-constrained optimization [73], etc., where models governed by PDEs need to be solved repeatedly at different samples of input variables.…”

Section: Operator Learning With Neural Networkmentioning

confidence: 99%

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

Cao¹,

O’Leary-Roseberry²,

Jha³

et al. 2022

Preprint

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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 a 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.

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“…The approximation quality also serves as the stopping criterion of the algorithm. Another approach involving physics-informed deep operator networks to accelerate PDE-constrained optimization in a self-supervised manner has recently been suggested in [33].…”

Section: Introductionmentioning

confidence: 99%

Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

et al. 2022

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In this contribution, we develop an efficient surrogate modeling framework for simulation-based optimization of enhanced oil recovery, where we particularly focus on polymer flooding. The computational approach is based on an adaptive training procedure of a neural network that directly approximates an input-output map of the underlying PDE-constrained optimization problem. The training process thereby focuses on the construction of an accurate surrogate model solely related to the optimization path of an outer iterative optimization loop. True evaluations of the objective function are used to finally obtain certified results. Numerical experiments are given to evaluate the accuracy and efficiency of the approach for a heterogeneous five-spot benchmark problem.

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Fast PDE-constrained optimization via self-supervised operator learning

Cited by 5 publications

References 43 publications

Deep Multi-Input and Multi-Output Operator Networks Method for Optimal Control of Pdes

Deep Multi-Input and Multi-Output Operator Networks Method for Optimal Control of Pdes

Residual-based error correction for neural operator accelerated infinite-dimensional Bayesian inverse problems

Adaptive machine learning-based surrogate modeling to accelerate PDE-constrained optimization in enhanced oil recovery

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