Numerical simulation for climate modeling resolving all important scales is a computationally taxing process. Therefore, to circumvent this issue a low resolution simulation is performed, which is subsequently corrected for bias using reanalyzed data (ERA5), known as nudging correction. The existing implementation for nudging correction uses a relaxation based method for the algebraic difference between low resolution and ERA5 data. In this study, we replace the bias correction process with a surrogate model based on the Deep Operator Network (DeepONet). DeepONet (Deep Operator Neural Network) learns the mapping from the state before nudging (a functional) to the nudging tendency (another functional). The nudging tendency is a very high dimensional data albeit having many low energy modes. Therefore, the DeepoNet is combined with a convolution based auto-encoder-decoder (AED) architecture in order to learn the nudging tendency in a lower dimensional latent space efficiently. The accuracy of the DeepONet model is tested against the nudging tendency obtained from the E3SMv2 (Energy Exascale Earth System Model) and shows good agreement. The overarching goal of this work is to deploy the DeepONet model in an online setting and replace the nudging module in the E3SM loop for better efficiency and accuracy.

Neural networks that solve regression problems map input data to output data, whereas neural operators map functions to functions. This recent paradigm shift in perspective, starting with the original paper on the deep operator network or DeepONet [1, 2], provides a new modeling capability that is very useful in engineering - that is, the ability to replace very complex and computational resource-taxing multiphysics systems with neural operators that can provide functional outputs in real-time. Specifically, unlike other physics-informed neural networks (PINNs) [3] that require optimization during training and testing, a DeepONet does not require any optimization during inference, hence it can be used in realtime forecasting, including design, autonomy, control, etc. An architectural diagram of a DeepONet with the commonly used nomenclature for its components is shown in Figure 1. DeepONets can take a multi-fidelity or multi-modal input [4, 5, 6, 7, 8] in the branch network and can use an independent network as the trunk, a network that represents the output space, e.g. in space-time coordinates or in parametric space in a continuous fashion. In some sense, DeepONets can be used as surrogates in a similar fashion as reduced order models (ROMs) [9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19]. However, unlike ROMs, they are over-parametrized which leads to both generalizability and robustness to noise that is not possible with ROMs, see the recent work of [20].

We introduce an optimized physics-informed neural network (PINN) trained to solve the problem of identifying and characterizing a surface breaking crack in a metal plate. PINNs are neural networks that can combine data and physics in the learning process by adding the residuals of a system of Partial Differential Equations to the loss function. Our PINN is supervised with realistic ultrasonic surface acoustic wave data acquired at a frequency of 5 MHz. The ultrasonic surface wave data is represented as a surface deformation on the top surface of a metal plate, measured by using the method of laser vibrometry. The PINN is physically informed by the acoustic wave equation and its convergence is sped up using adaptive activation functions. The adaptive activation function uses a scalable hyperparameter in the activation function, which is optimized to achieve best performance of the network as it changes dynamically the topology of the loss function involved in the optimization process. The usage of adaptive activation function significantly improves the convergence, notably observed in the current study. We use PINNs to estimate the speed of sound of the metal plate, which we do with an error of 1\%, and then, by allowing the speed of sound to be space dependent, we identify and characterize the crack as the positions where the speed of sound has decreased. Our study also shows the effect of sub-sampling of the data on the sensitivity of sound speed estimates. More broadly, the resulting model shows a promising deep neural network model for ill-posed inverse problems.