B-DeepONet: An enhanced Bayesian DeepONet for solving noisy parametric PDEs using accelerated replica exchange SGLD

Journal Article · Fri Oct 14 00:00:00 EDT 2022 · Journal of Computational Physics

DOI:https://doi.org/10.1016/j.jcp.2022.111713· OSTI ID:2421766

Lin, Guang ^[1]; Moya, Christian ^[2]; Zhang, Zecheng ^[2]

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1. Purdue Univ., West Lafayette, IN (United States); OSTI
2. Purdue Univ., West Lafayette, IN (United States)

Here, the Deep Operator Network (DeepONet) is a neural network architecture used to approximate operators, including the solution operator of parametric PDEs. DeepONets have shown remarkable approximation ability. However, the performance of DeepONets deteriorates when the training data is polluted with noise, a scenario that occurs in practice. To handle noisy data, we propose a Bayesian DeepONet based on replica exchange Langevin diffusion (reLD). Replica exchange uses two particles. The first particle trains a DeepONet to exploit the loss landscape and make predictions. The other particle trains a different DeepONet to explore the loss landscape and escape local minima via swapping. Compared to DeepONets trained with state-of-the-art gradient-based algorithms (e.g., Adam), the proposed Bayesian DeepONet greatly improves the training convergence for noisy scenarios and accurately estimates the uncertainty. To further reduce the high computational cost of the reLD training of DeepONets, we propose (1) an accelerated training framework that exploits the DeepONet's architecture to reduce its computational cost up to 25% without compromising performance and (2) a transfer learning strategy that accelerates training DeepONets for PDEs with different parameter values. Finally, we illustrate the effectiveness of the proposed Bayesian DeepONet using four parametric PDE problems.

• Research Organization: Brookhaven National Laboratory (BNL), Upton, NY (United States); Purdue Univ., West Lafayette, IN (United States)
• Sponsoring Organization: National Science Foundation (NSF); USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR)
• Grant/Contract Number: SC0021142; SC0023161
• OSTI ID: 2421766
• Alternate ID(s): OSTI ID: 1897743
• Journal Information: Journal of Computational Physics, Journal Name: Journal of Computational Physics Journal Issue: C Vol. 473; ISSN 0021-9991
• Publisher: Elsevier
• Country of Publication: United States
• Language: English

Related Subjects

97 MATHEMATICS AND COMPUTING
Bayesian method
Computer Science
Deep operator network
Langevin diffusion
Noisy data
Parametric PDE
Physics
Replica exchange