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Register a new userVariational Monte Carlo - Bridging Concepts of Machine Learning and High Dimensional Partial Differential Equations
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A statistical learning approach for parametric PDEs related to Uncertainty Quantification is derived. The method is based on the minimization of an empirical risk on a selected model class and it is shown to be applicable to a broad range of problems. A general unified convergence analysis is derived, which takes into account the approximation and the statistical errors. By this, a combination of theoretical results from numerical analysis and statistics is obtained. Numerical experiments illustrate the performance of the method with the model class of hierarchical tensors.
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High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number of financial assets in a portfolio. Moreover, such PDEs are often fully nonlinear due to the need to incorporate certain nonlinear phenomena in the model such as default risks, transaction costs, volatility uncertainty (Knightian uncertainty), or trading constraints in the model. Such high-dimensional fully nonlinear PDEs are exceedingly difficult to solve as the computational effort for standard approximation methods grows exponentially with the dimension. In this work we propose a new method for solving high-dimensional fully nonlinear second-order PDEs. Our method can in particular be used to sample from high-dimensional nonlinear expectations. The method is based on (i) a connection between fully nonlinear second-order PDEs and second-order backward stochastic differential equations (2BSDEs), (ii) a merged formulation of the PDE and the 2BSDE problem, (iii) a temporal forward discretization of the 2BSDE and a spatial approximation via deep neural nets, and (iv) a stochastic gradient descent-type optimization procedure. Numerical results obtained using ${rm T{small ENSOR}F{small LOW}}$ in ${rm P{small YTHON}}$ illustrate the efficiency and the accuracy of the method in the cases of a $100$-dimensional Black-Scholes-Barenblatt equation, a $100$-dimensional Hamilton-Jacobi-Bellman equation, and a nonlinear expectation of a $ 100 $-dimensional $ G $-Brownian motion.
The least squares method with deep neural networks as function parametrization has been applied to solve certain high-dimensional partial differential equations (PDEs) successfully; however, its convergence is slow and might not be guaranteed even within a simple class of PDEs. To improve the convergence of the network-based least squares model, we introduce a novel self-paced learning framework, SelectNet, which quantifies the difficulty of training samples, treats samples equally in the early stage of training, and slowly explores more challenging samples, e.g., samples with larger residual errors, mimicking the human cognitive process for more efficient learning. In particular, a selection network and the PDE solution network are trained simultaneously; the selection network adaptively weighting the training samples of the solution network achieving the goal of self-paced learning. Numerical examples indicate that the proposed SelectNet model outperforms existing models on the convergence speed and the convergence robustness, especially for low-regularity solutions.
In this paper we consider sequential joint state and static parameter estimation given discrete time observations associated to a partially observed stochastic partial differential equation (SPDE). It is assumed that one can only estimate the hidden state using a discretization of the model. In this context, it is known that the multi-index Monte Carlo (MIMC) method of [11] can be used to improve over direct Monte Carlo from the most precise discretizaton. However, in the context of interest, it cannot be directly applied, but rather must be used within another advanced method such as sequential Monte Carlo (SMC). We show how one can use the MIMC method by renormalizing the MI identity and approximating the resulting identity using the SMC$^2$ method of [5]. We prove that our approach can reduce the cost to obtain a given mean square error (MSE), relative to just using SMC$^2$ on the most precise discretization. We demonstrate this with some numerical examples.
Solving general high-dimensional partial differential equations (PDE) is a long-standing challenge in numerical mathematics. In this paper, we propose a novel approach to solve high-dimensional linear and nonlinear PDEs defined on arbitrary domains by leveraging their weak formulations. We convert the problem of finding the weak solution of PDEs into an operator norm minimization problem induced from the weak formulation. The weak solution and the test function in the weak formulation are then parameterized as the primal and adversarial networks respectively, which are alternately updated to approximate the optimal network parameter setting. Our approach, termed as the weak adversarial network (WAN), is fast, stable, and completely mesh-free, which is particularly suitable for high-dimensional PDEs defined on irregular domains where the classical numerical methods based on finite differences and finite elements suffer the issues of slow computation, instability and the curse of dimensionality. We apply our method to a variety of test problems with high-dimensional PDEs to demonstrate its promising performance.
As further progress in the accurate and efficient computation of coupled partial differential equations (PDEs) becomes increasingly difficult, it has become highly desired to develop new methods for such computation. In deviation from conventional approaches, this short communication paper explores a computational paradigm that couples numerical solutions of PDEs via machine-learning (ML) based methods, together with a preliminary study on the paradigm. Particularly, it solves PDEs in subdomains as in a conventional approach but develops and trains artificial neural networks (ANN) to couple the PDEs solutions at their interfaces, leading to solutions to the PDEs in the whole domains. The concepts and algorithms for the ML coupling are discussed using coupled Poisson equations and coupled advection-diffusion equations. Preliminary numerical examples illustrate the feasibility and performance of the ML coupling. Although preliminary, the results of this exploratory study indicate that the ML paradigm is promising and deserves further research.
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