BubbleOKAN: A physics-informed interpretable neural operator for high-frequency bubble dynamics

In this work, we employ physics-informed neural operators to map pressure profiles from an input function space to the corresponding bubble radius responses. Our approach employs a two-step DeepONet architecture. To address the intrinsic spectral bias of deep learning models, our model incorporates the Rowdy adaptive activation function, enhancing the representation of high-frequency features. Moreover, we introduce the Kolmogorov-Arnold network (KAN) based two-step DeepOKAN model, which enhances interpretability (often lacking in conventional multilayer perceptron architectures) while efficiently capturing high-frequency bubble dynamics without explicit utilization of activation functions in any form. We particularly investigate the use of spline basis functions in combination with radial basis functions (RBF) within our architecture, as they demonstrate superior performance in constructing a universal basis for approximating high-frequency bubble dynamics compared to alternative formulations. Furthermore, we emphasize on the performance bottleneck of RBF while learning the high frequency bubble dynamics and showcase the advantage of using spline basis function for the trunk network in overcoming this inherent spectral bias. The model is systematically evaluated across three representative scenarios: (1) bubble dynamics governed by the Rayleigh-Plesset equation with a single initial radius, (2) bubble dynamics governed by the Keller-Miksis equation with a single initial radius, and (3) Keller-Miksis dynamics with multiple initial radii. We also compare our results with state-of-the-art neural operators, including Fourier Neural Operators, Wavelet Neural Operators, OFormer, and Convolutional Neural Operators. Our findings demonstrate that the two-step DeepOKAN accurately captures both low- and high-frequency behaviors, and offers a promising alternative to conventional numerical solvers. The two-step DeepOKAN code is available at https://github.com/ParamIntelligence/Two-Step-DeepOKAN .