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Award ID contains: 2225507

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  1. ; ; ; ; (, IEEE)
    We present an efficient probabilistic solver for time-dependent nonlinear partial differential equations.We formulate our method as the maximum a posteriori solver for a constrained risk problem on a reproducing kernel Hilbert space induced by a spatio-temporal Gaussian process prior. We show that for a suitable choice of temporal kernels, the risk objective can be minimized efficiently via a Gauss–Newton algorithm cor- responding to an iterated extended Kalman smoother (IEKS). Furthermore, by leveraging a parallel-in-time implementation of IEKS, our algorithm can take advantage of massively parallel graphical processing units to achieve logarithmic instead of linear scaling with time. We validate our method numerically on popular benchmark problems. 
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  2. ; (, 1st workshop on Synergy of Scientific and Machine Learning Modeling, SynS & ML - ICML, Honolulu, Hawaii, USA. July, 2023.)
    PSO-PINN is a class of algorithms for training physics-informed neural networks (PINN) using particle swarm optimization (PSO). PSO-PINN can mitigate the well-known difficulties presented by gradient descent training of PINNs when dealing with PDEs with irregular solutions. Additionally, PSO-PINN is an ensemble approach to PINN that yields reproducible predictions with quantified uncertainty. In this paper, we introduce Multi-Objective PSO-PINN, which treats PINN training as a multi-objective problem. The proposed multi-objective PSO-PINN represents a new paradigm in PINN training, which thus far has relied on scalarizations of the multi-objective loss function. A full multi-objective approach allows on-the-fly compromises in the trade-off among the various components of the PINN loss function. Experimental results with a diffusion PDE problem demonstrate the promise of this methodology. 
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    Accepted Manuscript Full Text Available