BubbleOKAN: A Physics-Informed Interpretable Neural Operator for High-Frequency Bubble Dynamics

TL;DR

BubbleOKAN advances high-frequency bubble dynamics modeling by learning physics-informed operators that map pressure profiles to bubble radii using a two-step DeepONet and a Kolmogorov–Arnold Network–based DeepOKAN variant. Rowdy activations mitigate spectral bias in DeepONet, while continual learning and spline/RBF basis choices in DeepOKAN provide interpretable, efficient representations of complex, high-frequency bubble behavior under RP and KM dynamics. The framework is validated across single and multi-radius scenarios and benchmarked against state-of-the-art neural operators, demonstrating accurate reproduction of both low- and high-frequency content with strong generalization in many but not all extrapolation regimes. Limitations include resonance-frequency cases and out-of-distribution predictions, guiding future work toward enhanced basis selection, broader frequency coverage, and real-time multi-bubble applications.

Abstract

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.

Figures (21)

• Figure 1: A bubble dynamics simulation based on R-P equation obtained from APECSS for a bubble with $R_0 = 50\mu m, f = 1970 \text{KHz}, ~\text{amp} = 3\times10^5Pa, t = 50\mu s$. Top: pressure profile. Bottom: Radius evolution
• Figure 2: Validation results from R-P equation using the vanilla DeepONet after 500,000 epochs. (a) Bubble dynamics driven by pressure with frequency of 676 KHz and amplitude of $3 \times 10^5\,\text{Pa}$, representing low frequency case. (b) Bubble dynamics driven by pressure with frequency of 1654 KHz and amplitude of $10 \times 10^5\,\text{Pa}$, representing high frequency case.
• Figure 3: Architecture of DeepONet: The operator layer employs a Kronecker-based branch net (input: full pressure profile) and a trunk net (input: time and initial radius). The operation layer enforces physical constraints on outputs, while training minimizes a combined data-ODE loss via iterative optimization.
• Figure 4: Two step training architecture for DeepONet.
• Figure 5: Two step training architecture for DeepOKAN.