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Learning and meta-learning of stochastic advection–diffusion–reaction systems from sparse measurements

Part of: Artificial intelligence (68Txx)

Published online by Cambridge University Press:  15 June 2020

XIAOLI CHEN Open the ORCID record for XIAOLI CHEN [Opens in a new window] ,
JINQIAO DUAN Open the ORCID record for JINQIAO DUAN [Opens in a new window]  and
GEORGE EM KARNIADAKIS Open the ORCID record for GEORGE EM KARNIADAKIS [Opens in a new window]
XIAOLI CHEN
Affiliation:
Center for Mathematical Sciences and School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan430074, China email: xlchen@hust.edu.cn Division of Applied Mathematics, Brown University, Providence, RI02912, USA email: george_karniadakis@brown.edu
JINQIAO DUAN
Affiliation:
Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL60616, USA email: duan@iit.edu
GEORGE EM KARNIADAKIS
Affiliation:
Division of Applied Mathematics, Brown University, Providence, RI02912, USA email: george_karniadakis@brown.edu Pacific Northwest National Laboratory, Richland, WA99354, USA
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Abstract

Physics-informed neural networks (PINNs) were recently proposed in [18] as an alternative way to solve partial differential equations (PDEs). A neural network (NN) represents the solution, while a PDE-induced NN is coupled to the solution NN, and all differential operators are treated using automatic differentiation. Here, we first employ the standard PINN and a stochastic version, sPINN, to solve forward and inverse problems governed by a non-linear advection–diffusion–reaction (ADR) equation, assuming we have some sparse measurements of the concentration field at random or pre-selected locations. Subsequently, we attempt to optimise the hyper-parameters of sPINN by using the Bayesian optimisation method (meta-learning) and compare the results with the empirically selected hyper-parameters of sPINN. In particular, for the first part in solving the inverse deterministic ADR, we assume that we only have a few high-fidelity measurements, whereas the rest of the data is of lower fidelity. Hence, the PINN is trained using a composite multi-fidelity network, first introduced in [12], that learns the correlations between the multi-fidelity data and predicts the unknown values of diffusivity, transport velocity and two reaction constants as well as the concentration field. For the stochastic ADR, we employ a Karhunen–Loève (KL) expansion to represent the stochastic diffusivity, and arbitrary polynomial chaos (aPC) to represent the stochastic solution. Correspondingly, we design multiple NNs to represent the mean of the solution and learn each aPC mode separately, whereas we employ a separate NN to represent the mean of diffusivity and another NN to learn all modes of the KL expansion. For the inverse problem, in addition to stochastic diffusivity and concentration fields, we also aim to obtain the (unknown) deterministic values of transport velocity and reaction constants. The available data correspond to 7spatial points for the diffusivity and 20 space–time points for the solution, both sampled 2000 times. We obtain good accuracy for the deterministic parameters of the order of 1–2% and excellent accuracy for the mean and variance of the stochastic fields, better than three digits of accuracy. In the second part, we consider the previous stochastic inverse problem, and we use Bayesian optimisation to find five hyper-parameters of sPINN, namely the width, depth and learning rate of two NNs for learning the modes. We obtain much deeper and wider optimal NNs compared to the manual tuning, leading to even better accuracy, i.e., errors less than 1% for the deterministic values, and about an order of magnitude less for the stochastic fields.

Keywords

Physics-informed neural networksarbitrary polynomial chaosmulti-fidelity dataKarhunen–Loève expansionuncertainty quantificationBayesian optimisationinverse problems

MSC classification

Primary: 68T05: Learning and adaptive systems
Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 3: Connections between Deep learning and Partial Differential Equations , June 2021 , pp. 397 - 420
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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