PDE Constraints Research Articles

High-dimensional inverse modeling of hydraulic tomography by physics informed neural network (HT-PINN)

A hydraulic tomography – physics informed neural network (HT-PINN) is developed for inverting two-dimensional large-scale spatially distributed transmissivity. HT-PINN involves a neural network model of transmissivity and a series of neural network models to describe transient or steady-state sequential pumping tests. All the neural network models are jointly trained by minimizing the total loss function including data fitting errors and PDE constraints. Batch training of collocation points is used to amplify the advantage of the mesh-free property of neural networks, thereby limiting the number of collocation points per training iteration and reducing the total training time. The developed HT-PINN accurately and efficiently inverts two-dimensional Gaussian transmissivity fields with more than a million unknowns (1024 × 1024 resolution), and the inversion map accuracy exceeds 95 %. The effects of batch sampling methods, batch number and size, and data requirements for direct and indirect measurements are systematically investigated. In addition, the developed HT-PINN exhibits great scalability and structure robustness in inverting fields with different resolutions ranging from coarse (64 × 64) to fine (1024 × 1024). Specifically, data requirements do not increase with the problem dimensionality, and the computational cost of HT-PINN remains almost unchanged due to its mesh-free nature while maintaining high inversion accuracy when increasing the field resolution.

PA_CasualLSTM: A new time series prediction network with the physical constraint and adjusted Fourier neural operator for the time-dependent partial differential equation

In this work, a new time series prediction network is proposed in the framework of CasualLSTM with physical constraints and an adjusted Fourier neural operator (FNO) for the solution of the time-dependent partial differential equation. The framework of CasualLSTM is employed to learn the time evolution of spatial features which strengthens the extrapolation capability. With the help of adjusted Fourier layers (AFLs), residual connection, and the adaptive time-marching strategy, the network can quickly converge and extrapolate without labeled data by encoding PDE constraints into loss functions. Two examples, namely, Burger’s equation and two-dimensional Navier–Stokes (N-S) equation are used to evaluate the proposed method. Numerical results show that the proposed method has a good performance in solution accuracy and extrapolability.

Solving PDE-Constrained Control Problems Using Operator Learning

The modeling and control of complex physical systems are essential in real-world problems. We propose a novel framework that is generally applicable to solving PDE-constrained optimal control problems by introducing surrogate models for PDE solution operators with special regularizers. The procedure of the proposed framework is divided into two phases: solution operator learning for PDE constraints (Phase 1) and searching for optimal control (Phase 2). Once the surrogate model is trained in Phase 1, the optimal control can be inferred in Phase 2 without intensive computations. Our framework can be applied to both data-driven and data-free cases. We demonstrate the successful application of our method to various optimal control problems for different control variables with diverse PDE constraints from the Poisson equation to Burgers' equation.

Numerical approximation of optimal convex and rotationally symmetric shapes for an eigenvalue problem arising in optimal insulation

We are interested in the optimization of convex domains under a PDE constraint. Due to the difficulties of approximating convex domains in R3, the restriction to rotationally symmetric domains is used to reduce shape optimization problems to a two-dimensional setting. For the optimization of an eigenvalue arising in a problem of optimal insulation, the existence of an optimal domain is proven. An algorithm is proposed that can be applied to general shape optimization problems under the geometric constraints of convexity and rotational symmetry. The approximated optimal domains for the eigenvalue problem in optimal insulation are discussed.

Determining kernels in linear viscoelasticity

In this work, we investigate the inverse problem of determining the kernel functions that best describe the mechanical behavior of a complex medium modeled by a general nonlocal viscoelastic wave equation. To this end, we minimize a tracking-type data misfit function under this PDE constraint. We perform the well-posedness analysis of the state and adjoint problems and, using these results, rigorously derive the first-order sensitivities. Numerical experiments in a three-dimensional setting illustrate the method.

Optimal convergence of finite element approximation to an optimization problem with PDE constraint* *Wei Gong was supported by the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDB 41000000), the National Key Basic Research Program (Grant No. 2018YFB0704304) and the National Natural Science Foundation of China (Grant No. 12071468, 11671391). Zhaojie Zhou was supported by the National

We study in this paper the optimal convergence of finite element approximation to an optimization problem with PDE constraint. Specifically, we consider an elliptic distributed optimal control problem without control constraints, which can also be viewed as a regularized inverse source problem. The main contributions are two-fold. First, we derive a priori and a posteriori error estimates for the optimization problems, under an appropriately chosen norm that allows us to establish an isomorphism between the solution space and its dual. These results yield error estimates with explicit dependence on the regularization parameter α so that the constants appeared in the derivation are independent of α. Second, we prove the contraction property and rate optimality for the adaptive algorithm with respect to the error estimator and solution errors between the adaptive finite element solutions and the continuous solutions. Extensive numerical experiments are presented that confirm our theoretical results.

Spectral, Tensor and Domain Decomposition Methods for Fractional PDEs

Abstract Fractional PDEs have recently found several geophysics and imaging science applications due to their nonlocal nature and their flexibility in capturing sharp transitions across interfaces. However, this nonlocality makes it challenging to design efficient solvers for such problems. In this paper, we introduce a spectral method based on an ultraspherical polynomial discretization of the Caffarelli–Silvestre extension to solve such PDEs on rectangular and disk domains. We solve the discretized problem using tensor equation solvers and thus can solve higher-dimensional PDEs. In addition, we introduce both serial and parallel domain decomposition solvers. We demonstrate the numerical performance of our methods on a 3D fractional elliptic PDE on a cube as well as an application to optimization problems with fractional PDE constraints.

PDE Constrained Inverse Problems arising from Turbulence Modeling

AbstractTurbulence models, which are a means to fix the closure problem arising from Reynolds averaging of Navier‐Stokes equations, are economical stop‐gaps but suffer from accuracy issues. Modifying turbulence models by incorporating corrections in their functional form is one approach to improve their accuracy. We estimate correction functionals for the Spalart ‐ Allmaras turbulence model, based on an inverse problem with PDE constraints emphasizing the issue of regularization.

Comparison of Shape Derivatives Using CutFEM for Ill-posed Bernoulli Free Boundary Problem

In this paper we study and compare three types of shape derivatives for free boundary identification problems. The problem takes the form of a severely ill-posed Bernoulli problem where only the Dirichlet condition is given on the free (unknown) boundary, whereas both Dirichlet and Neumann conditions are available on the fixed (known) boundary. Our framework resembles the classical shape optimization method in which a shape dependent cost functional is minimized among the set of admissible domains. The position of the domain is defined implicitly by the level set function. The steepest descent method, based on the shape derivative, is applied for the level set evolution. For the numerical computation of the gradient, we apply the Cut Finite Element Method (CutFEM), that circumvents meshing and re-meshing, without loss of accuracy in the approximations of the involving partial differential models. We consider three different shape derivatives. The first one is the classical shape derivative based on the cost functional with pde constraints defined on the continuous level. The second shape derivative is similar but using a discretized cost functional that allows for the embedding of CutFEM formulations directly in the formulation. Different from the first two methods, the third shape derivative is based on a discrete formulation where perturbations of the domain are built into the variational formulation on the unperturbed domain. This is realized by using the so-called boundary value correction method that was originally introduced to allow for high order approximations to be realized using low order approximation of the domain. The theoretical discussion is illustrated with a series of numerical examples showing that all three approaches produce similar result on the proposed Bernoulli problem.

A non-conforming dual approach for adaptive Trust-Region reduced basis approximation of PDE-constrained parameter optimization

In this contribution we propose and rigorously analyze new variants of adaptive Trust-Region methods for parameter optimization with PDE constraints and bilateral parameter constraints. The approach employs successively enriched Reduced Basis surrogate models that are constructed during the outer optimization loop and used as model function for the Trust-Region method. Each Trust-Region sub-problem is solved with the projected BFGS method. Moreover, we propose a non-conforming dual (NCD) approach to improve the standard RB approximation of the optimality system. Rigorous improved a posteriori error bounds are derived and used to prove convergence of the resulting NCD-corrected adaptive Trust-Region Reduced Basis algorithm. Numerical experiments demonstrate that this approach enables to reduce the computational demand for large scale or multi-scale PDE constrained optimization problems significantly.

An efficient method for optimizing the unsteady heat and mass transport features for convective drying of two porous moist objects in a channel

In the present study, an efficient methodology is proposed for optimizing the convective drying performance of multiple porous moist objects in a channel. As the first step of the method, a PDE constraint optimization routine is utilized to obtain the optimum values of distance between the objects for maximizing the transfer rates while in the second part, unsteady coupled field equations for the channel and porous moist object domains are invoked for convective drying of multiple objects for the optimum spacing values. Finite element method is used for the numerical simulations. It was observed that the recirculation zones established behind the rectangular objects are profoundly affected with the distance between the objects. There is 12.5% variation in the average Nu for the first block with varying the horizontal distance between the objects. The effects of vertical spacing on the average Nu is profound for the second object. while up to 80% enhancement in the average Nu is obtained when the value is changed from sy = 0 to sy = 1.5hp. However, the optimum values of the distance that maximize the heat transfer are obtained as sx = 1.1hp and sy = 1.931hp for the horizontal distance and vertical distance. At this optimum values of parameters, reduction in the moisture content becomes 33.4% and 98.015% for drying times at 1000 s and 5000 s. These values are also checked with the parametric unsteady coupled field equations for the porous moist objects which shows the improved time dependent drying features for the two blocks at the optimum points.

Spectral analysis of saddle-point matrices from optimization problems with elliptic PDE constraints

The main focus of this paper is the characterization and exploitation of the asymptotic spectrum of the saddle--point matrix sequences arising from the discretization of optimization problems constrained by elliptic partial differential equations. They uncover the existence of an hidden structure in these matrix sequences, namely, they show that these are indeed an example of Generalized Locally Toeplitz (GLT) sequences. They show that this enables a sharper characterization of the spectral properties of such sequences than the one that is available by using only the fact that they deal with saddle--point matrices. Finally, they exploit it to propose an optimal preconditioner strategy for the GMRES, and Flexible--GMRES methods.

On Regularization of an Optimal Control Problem for Ill-Posed Nonlinear Elliptic Equations

We discuss the existence issue to an optimal control problem for one class of nonlinear elliptic equations with an exponential type of nonlinearity. We deal with the control object when we cannot expect to have a solution of the corresponding boundary value problem in the standard functional space for all admissible controls. To overcome this difficulty, we make use of a variant of the classical Tikhonov regularization scheme. In particular, we eliminate the PDE constraints between control and state and allow such pairs run freely by introducing an additional variable which plays the role of “compensator” that appears in the original state equation. We show that this fictitious variable can be determined in a unique way. In order to provide an approximation of the original optimal control problem, we define a special family of regularized optimization problems. We show that each of these problems is consistent, well-posed, and their solutions allow to attain an optimal solution of the original problem as the parameter of regularization tends to zero. As a consequence, we prove the existence of optimal solutions to the original problem and propose a way for their approximation.

Unconditional convergence for discretizations of dynamical optimal transport

The dynamical formulation of optimal transport, also known as Benamou-Brenier formulation or Computational Fluid Dynamic formulation, amounts to write the optimal transport problem as the optimization of a convex functional under a PDE constraint, and can handle a priori a vast class of cost functions and geometries. Several disretizations of this problem have been proposed, leading to computations on flat spaces as well as Riemannian manifolds, with extensions to mean field games and gradient flows in the Wasserstein space. In this article, we provide a framework which guarantees convergence under mesh refinement of the solutions of the space-time discretized problems to the one of the infinite-dimensional one for quadratic optimal transport. The convergence holds without condition on the ratio between spatial and temporal step sizes, and can handle arbitrary positive measures as input, while the underlying space can be a Riemannian manifold. Both the finite volume discretization proposed by Gladbach, Kopfer and Maas, as well as the discretization over triangulations of surfaces studied by the present author in collaboration with Claici, Chien and Solomon fit in this framework.

Efficiency in uncertain variational control problems

In this paper, considering the applications of interval analysis in various fields (such as artificial intelligence, neural computation, genetic algorithms, information theory or fuzzy logic), a new class of interval-valued variational control problems governed by multiple integral functionals, first-order PDE and inequality constraints is studied. More precisely, efficiency conditions for the considered uncertain variational control problem are formulated and proved. The sufficiency of Karush–Kuhn–Tucker conditions is established under some invexity and $$(\rho, b)$$ -quasiinvexity assumptions of the involved functionals. In addition, the paper is completed with illustrative applications (describing the controlled behavior of an artificial neural system) and the corresponding algorithm.

Multigrid preconditioners for optimal control problems with stochastic elliptic PDE constraints

ABSTRACT In this work, we construct multigrid preconditioners to be used in the solution process of pathwise optimal control problems constrained by elliptic partial differential equations with random coefficients. We use a sparse-grid collocation method to discretize in the stochastic space and multigrid techniques in the physical space. Numerical results show that the proposed preconditioners lead to significant computational savings, with the number of preconditioned conjugate gradient iterations decreasing as the resolution in the physical space increases.

Deep learning of subsurface flow via theory-guided neural network

Active researches are currently being performed to incorporate the wealth of scientific knowledge into data-driven approaches (e.g., neural networks) in order to improve the latter’s effectiveness. In this study, the Theory-guided Neural Network (TgNN) is proposed for deep learning of subsurface flow. In the TgNN, as supervised learning, the neural network is trained with available observations or simulation data while being simultaneously guided by theory (e.g., governing equations, other physical constraints, engineering controls, and expert knowledge) of the underlying problem. The TgNN can achieve higher accuracy than the ordinary Deep Neural Network (DNN) because the former provides physically feasible predictions and can be more readily generalized beyond the regimes covered with the training data. Furthermore, the TgNN model is proposed for subsurface flow with heterogeneous model parameters. Several numerical cases of two-dimensional transient saturated flow are introduced to test the performance of the TgNN. In the learning process, the loss function contains data mismatch, as well as PDE constraint, engineering control, and expert knowledge. After obtaining the parameters of the neural network by minimizing the loss function, a TgNN model is built that not only fits the data, but also adheres to physical/engineering constraints. Predicting the future response can be easily realized by the TgNN model. In addition, the TgNN model is tested in more complicated scenarios, such as prediction with changed boundary conditions, learning from noisy data or outliers, transfer learning, learning from sparse data, and engineering controls. Numerical results demonstrate that the TgNN model achieves much better predictability, reliability, and generalizability than DNN models due to the physical/engineering constraints in the former.

Multigoal-oriented optimal control problems with nonlinear PDE constraints

In this work, we consider an optimal control problem subject to a nonlinear PDE constraint and apply it to a semi-linear monotone PDE and the regularized p-Laplace equation. To this end, a reduced unconstrained optimization problem in terms of the control variable is formulated. Based on the reduced approach, we then derive an a posteriori error representation and mesh adaptivity for multiple quantities of interest. All quantities are combined to one, and then the dual-weighted residual (DWR) method is applied to this combined functional. Furthermore, the estimator allows for balancing the discretization error and the nonlinear iteration error. These developments allow us to formulate an adaptive solution strategy, which is finally substantiated with the help of several numerical examples.

Quasi-best approximation in optimization with PDE constraints

We consider finite element solutions to quadratic optimization problems, where the state depends on the control via a well-posed linear partial differential equation. Exploiting the structure of a suitably reduced optimality system, we prove that the combined error in the state and adjoint state of the variational discretization is bounded by the best approximation error in the underlying discrete spaces. The constant in this bound depends on the inverse square root of the Tikhonov regularization parameter. Furthermore, if the operators of control action and observation are compact, this quasi-best approximation constant becomes independent of the Tikhonov parameter as the mesh size tends to 0 and we give quantitative relationships between mesh size and Tikhonov parameter ensuring this independence. We also derive generalizations of these results when the control variable is discretized or when it is taken from a convex set.

Numerical Approximation of Optimal Convex Shapes

This article investigates the numerical approximation of shape optimization problems with PDE constraint on classes of convex domains. The convexity constraint provides a compactness property which...