
Learning in Modal Space: Solving TimeDependent Stochastic PDEs Using PhysicsInformed Neural Networks
One of the open problems in scientific computing is the longtime integration of nonlinear stochastic partial differential equations (SPDEs). We address this problem by taking advantage of recent advances in scientific machine learning and the dynamically orthogonal (DO) and biorthogonal (BO) methods for representing stochastic processes. Specifically, we propose two new PhysicsInformed Neural Networks (PINNs) for solving timedependent SPDEs, namely the NNDO/BO methods, which incorporate the DO/BO constraints into the loss function with an implicit form instead of generating explicit expressions for the temporal derivatives of the DO/BO modes. Hence, the proposed methods overcome some of the drawbacks of the original DO/BO methods: we do not need the assumption that the covariance matrix of the random coefficients is invertible as in the original DO method, and we can remove the assumption of no eigenvalue crossing as in the original BO method. Moreover, the NNDO/BO methods can be used to solve timedependent stochastic inverse problems with the same formulation and computational complexity as for forward problems. We demonstrate the capability of the proposed methods via several numerical examples: (1) A linear stochastic advection equation with deterministic initial condition where the original DO/BO method would fail; (2) Longtime integration of the stochastic Burgers' equation with many eigenvalue crossings during the whole time evolution where the original BO method fails. (3) Nonlinear reaction diffusion equation: we consider both the forward and the inverse problem, including noisy initial data, to investigate the flexibility of the NNDO/BO methods in handling inverse and mixed type problems. Taken together, these simulation results demonstrate that the NNDO/BO methods can be employed to effectively quantify uncertainty propagation in a wide range of physical problems.
05/03/2019 ∙ by Dongkun Zhang, et al. ∙ 6 ∙ shareread it

Quantifying total uncertainty in physicsinformed neural networks for solving forward and inverse stochastic problems
Physicsinformed neural networks (PINNs) have recently emerged as an alternative way of solving partial differential equations (PDEs) without the need of building elaborate grids, instead, using a straightforward implementation. In particular, in addition to the deep neural network (DNN) for the solution, a second DNN is considered that represents the residual of the PDE. The residual is then combined with the mismatch in the given data of the solution in order to formulate the loss function. This framework is effective but is lacking uncertainty quantification of the solution due to the inherent randomness in the data or due to the approximation limitations of the DNN architecture. Here, we propose a new method with the objective of endowing the DNN with uncertainty quantification for both sources of uncertainty, i.e., the parametric uncertainty and the approximation uncertainty. We first account for the parametric uncertainty when the parameter in the differential equation is represented as a stochastic process. Multiple DNNs are designed to learn the modal functions of the arbitrary polynomial chaos (aPC) expansion of its solution by using stochastic data from sparse sensors. We can then make predictions from new sensor measurements very efficiently with the trained DNNs. Moreover, we employ dropout to correct the overfitting and also to quantify the uncertainty of DNNs in approximating the modal functions. We then design an active learning strategy based on the dropout uncertainty to place new sensors in the domain to improve the predictions of DNNs. Several numerical tests are conducted for both the forward and the inverse problems to quantify the effectiveness of PINNs combined with uncertainty quantification. This NNaPC new paradigm of physicsinformed deep learning with uncertainty quantification can be readily applied to other types of stochastic PDEs in multidimensions.
09/21/2018 ∙ by Dongkun Zhang, et al. ∙ 0 ∙ shareread it

I4U Submission to NIST SRE 2018: Leveraging from a Decade of Shared Experiences
The I4U consortium was established to facilitate a joint entry to NIST speaker recognition evaluations (SRE). The latest edition of such joint submission was in SRE 2018, in which the I4U submission was among the bestperforming systems. SRE'18 also marks the 10year anniversary of I4U consortium into NIST SRE series of evaluation. The primary objective of the current paper is to summarize the results and lessons learned based on the twelve subsystems and their fusion submitted to SRE'18. It is also our intention to present a shared view on the advancements, progresses, and major paradigm shifts that we have witnessed as an SRE participant in the past decade from SRE'08 to SRE'18. In this regard, we have seen, among others, a paradigm shift from supervector representation to deep speaker embedding, and a switch of research challenge from channel compensation to domain adaptation.
04/16/2019 ∙ by Kong Aik Lee, et al. ∙ 0 ∙ shareread it
Ling Guo
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