Solutions Of Stochastic Partial Differential Equations Research Articles

Deep Latent Regularity Network for Modeling Stochastic Partial Differential Equations

Stochastic partial differential equations (SPDEs) are crucial for modelling dynamics with randomness in many areas including economics, physics, and atmospheric sciences. Recently, using deep learning approaches to learn the PDE solution for accelerating PDE simulation becomes increasingly popular. However, SPDEs have two unique properties that require new design on the models. First, the model to approximate the solution of SPDE should be generalizable over both initial conditions and the random sampled forcing term. Second, the random forcing terms usually have poor regularity whose statistics may diverge (e.g., the space-time white noise). To deal with the problems, in this work, we design a deep neural network called \emph{Deep Latent Regularity Net} (DLR-Net). DLR-Net includes a regularity feature block as the main component, which maps the initial condition and the random forcing term to a set of regularity features. The processing of regularity features is inspired by regularity structure theory and the features provably compose a set of basis to represent the SPDE solution. The regularity features are then fed into a small backbone neural operator to get the output. We conduct experiments on various SPDEs including the dynamic $\Phi^4_1$ model and the stochastic 2D Navier-Stokes equation to predict their solutions, and the results demonstrate that the proposed DLR-Net can achieve SOTA accuracy compared with the baselines. Moreover, the inference time is over 20 times faster than the traditional numerical solver and is comparable with the baseline deep learning models.

A deep neural network method for solving partial differential equations with complex boundary in groundwater seepage

High dimensional stochastic partial differential equations (SPDEs) attract a lot of attention because of their application in uncertainty quantification (UQ) of ore deposits, petroleum geology and other fields. The groundwater flow in heterogeneous media affects all geological processes, including diagenesis, mineralization and oil accumulation. A point of the deep neural network (DNN) method in machine learning is that it can effectively output the solution of the SPDEs with the complex boundaries such as Newman boundary in a direct way. We propose a novel methodology to construct approximate solutions for SPDEs in groundwater flow by combining two deep convolutional residual networks, which can complete the training without interference in the help of an adaptive functional factor. One of the two network models deals with the inner observed points and the other one is required to satisfy the boundary conditions. This model has great potential for solving more complex boundary problems. Another contribution of this work is that the loss function to train the proposed DNN is obtained by deducing the energy functional based on the variational principles, which integrates the physical meaning into the proposed DNN. In addition, we theoretically prove that the minimization loss function problem has a unique solution, and it always conforms to the weak solution form of the problem no matter how the uncertain parameters change. An unconstrained optimization problem, rather than a constrained problem, can be solved without adding any penalty terms. The practical application shows that our model has low computational cost and strong efficiency for solving high-dimensional SPDEs.

Maximum principle for optimal control of SPDEs with locally monotone coefficients

ABSTRACT The aim of this paper is to derive a maximum principle for a control problem governed by a stochastic partial differential equation (SPDE) with locally monotone coefficients. To reach our goal we adapt the method which uses the relation between backward stochastic partial differential equation (BSPDE) and the maximum principle. In particular, necessary conditions for optimality for this stochastic optimal control problem are obtained. In spite of the fact that the method used here was used by several authors before, our adaptation is not immediate. It applies a trick which is used to get estimates for solutions of SPDE with Locally Monotone Coefficients as in the proof of the Lemmas 5.1 and 5.3. This adaptation permits us to apply our results to get a maximum principle for the optimal control to the cases when the system is governed by the 2D stochastic Navier-Stokes equation and by a stochastic Burgers' equation.

Exponential moments for numerical approximations of stochastic partial differential equations

Stochastic partial differential equations (SPDEs) have become a crucial ingredient in a number of models from economics and the natural sciences. Many SPDEs that appear in such applications include non-globally monotone nonlinearities. Solutions of SPDEs with non-globally monotone nonlinearities are in nearly all cases not known explicitly. Such SPDEs can thus only be solved approximatively and it is an important research problem to construct and analyze discrete numerical approximation schemes which converge with positive strong convergence rates to the solutions of such infinite dimensional SPDEs. In the case of finite dimensional stochastic ordinary differential equations (SODEs) with non-globally monotone nonlinearities it has recently been revealed that exponential integrability properties of the discrete numerical approximation scheme are a key instrument to establish positive strong convergence rates for the considered approximation scheme. To the best of our knowledge, there exists no result in the scientific literature which proves exponential integrability properties for a time discrete approximation scheme in the case of an infinite dimensional SPDE. In this paper we propose a new class of tamed space-time-noise discrete exponential Euler approximation schemes that admit exponential integrability properties in the case of infinite dimensional SPDEs. In particular, we establish exponential moment bounds for the proposed approximation schemes in the case of stochastic Burgers equations, stochastic Kuramoto-Sivashinsky equations, and two-dimensional stochastic Navier-Stokes equations.

Malliavin Differentiability of Solutions of SPDEs with Lévy White Noise

We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term σ. We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If σ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.

Pathwise nonuniqueness for the SPDEs of some super-Brownian motions with immigration

We prove pathwise nonuniqueness in the stochastic partial differential equations (SPDEs) for some one-dimensional super-Brownian motions with immigration. In contrast to a closely related case investigated by Mueller, Mytnik and Perkins [Ann. Probab. 42 (2014) 2032-2112], the solutions of the present SPDEs are assumed to be nonnegative and have very different properties including uniqueness in law. In proving possible separation of solutions, we derive delicate properties of certain correlated approximating solutions, which is based on a novel coupling method called continuous decomposition. In general, this method may be of independent interest in furnishing solutions of SPDEs with intrinsic adapted structure.oximating solutions, which may be of independent interest in the study of superprocesses with immigration.

Numerical solution of stochastic partial differential equations using a collocation method

In this article we apply spectral collocation method to find a numerical solution of stochastic partial differential equations (SPDEs). Spectral collocation method is known to be impressively efficient for PDEs. We investigate this method for numerical solution of SPDEs and we obtain its rate of convergence. At first, the results are expressed for equations with globally Lipschitz coefficient, then we extend it to cases with locally Lipschitz coefficient. The analysis is supported by numerical results for some important SPDEs such as stochastic Kuramoto-Sivashinksy equation.

Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise

Motivated by applications to various semilinear and quasi-linear stochastic partial differential equations (SPDEs) appeared in real world models, we establish the existence and uniqueness of strong solutions to coercive and locally monotone SPDEs driven by Lévy processes. We illustrate the main results of our paper by showing how it can be applied to a large class of SPDEs such as stochastic reaction–diffusion equations, stochastic Burgers type equations, stochastic 2D hydrodynamical systems and stochastic equations of non-Newtonian fluids, which generalize many existing results in the literature.

Stochastic Collocation for Optimal Control Problems with Stochastic PDE Constraints

We discuss the use of stochastic collocation for the solution of optimal control problems which are constrained by stochastic partial differential equations (SPDE). Thereby the constraining, SPDE depends on data which is not deterministic but random. Assuming a deterministic control, randomness within the states of the input data will propagate to the states of the system. For the solution of SPDEs there has recently been an increasing effort in the development of efficient numerical schemes based upon the mathematical concept of generalized polynomial chaos. Modal-based stochastic Galerkin and nodal-based stochastic collocation versions of this methodology exist, both of which rely on a certain level of smoothness of the solution in the random space to yield accelerated convergence rates. In this paper we apply the stochastic collocation method to develop a gradient descent as well as a sequential quadratic program (SQP) for the minimization of objective functions constrained by an SPDE. The stochastic function involves several higher-order moments of the random states of the system as well as classical regularization of the control. In particular we discuss several objective functions of tracking type. Numerical examples are presented to demonstrate the performance of our new stochastic collocation minimization approach.

Stochastic viscosity solutions for SPDEs with continuous coefficients

In this note, nonlinear stochastic partial differential equations (SPDEs) with continuous coefficients are studied. Via the solutions of backward doubly stochastic differential equations (BDSDEs) with continuous coefficients, we provide an existence result of stochastic viscosity sub- and super-solutions to this class of SPDEs. Under some stronger conditions, we prove the existence of stochastic viscosity solutions.

Stochastic model reduction for chaos representations

This paper addresses issues of model reduction of stochastic representations and computational efficiency of spectral stochastic Galerkin schemes for the solution of partial differential equations with stochastic coefficients. In particular, an algorithm is developed for the efficient characterization of a lower dimensional manifold occupied by the solution to a stochastic partial differential equation (SPDE) in the Hilbert space spanned by Wiener chaos. A description of the stochastic aspect of the problem on two well-separated scales is developed to enable the stochastic characterization on the fine scale using algebraic operations on the coarse scale. With such algorithms at hand, the solution of SPDE’s becomes both computationally manageable and efficient. Moreover, a solid foundation is thus provided for the adaptive error control in stochastic Galerkin procedures. Different aspects of the proposed methodology are clarified through its application to an example problem from solid mechanics.

Advanced Topics in Computational Partial Differential Equations: Numerical Methods and Diffpack Programming

Advanced Topics in Computational Partial Differential Equations: Numerical Methods and Diffpack Programming