aiqbench analysis
This paper applies Physics-Informed Neural Networks, with a sinusoidal activation, to the nonlinear Westervelt equation for finite-amplitude ultrasound in tissue, and reports residuals and losses converging across linear, shock-dominated, and dissipation-dominated test cases. The results are on synthetic parameter sets, so the practical value for real tissue imaging is still unproven. For ML readers, it is a useful data point on PINNs for stiff, oscillatory nonlinear PDEs, where the open questions are how the method compares with conventional solvers and how it scales.
Solving the Nonlinear Westervelt Equation in Ultrasound Propagation Using Physics-Informed Neural Networks
arXiv:2611.05130v1 Announce Type: cross Abstract: Modeling the nonlinear acoustic wave propagation in biological tissues needs accuracy, especially in ultrasound imaging and therapeutic applications. The 1D Westervelt equation, \( p_{xx} - p_{tt} + \gamma\, p_{ttt} + \eta\,(p^{2})_{tt}=0, \) could be used to describe a finite-amplitude ultrasound propagation in these tissues. This equation accounts for nonlinearity, \(\eta\), and attenuation, \(\gamma\). To solve this equation, we observe that traditional solvers may struggle to do so correctly. This is due to the difficulties arising from the nonlinear nature of the equation. We solve the equation by defining three synthetic cases - linear (\(\eta=0.001\), \(\gamma=10^{-6}\)), shock-dominated (\(\eta=0.02\), \(\gamma=10^{-6}\)), and dissipation-dominated (\(\eta=0.01\), \(\gamma=5\times10^{-4}\)) - and using Physics-Informed Neural Networks (PINNs) to showcase the solution. A sinusoidal activation function is used to capture the oscillatory behavior of the problem. The results of our study, with PDE residual converging to at least \(10^{-6}\), boundary condition loss to \(10^{-5}\), and the PINN loss converging to \(10^{-5}\) for each case at \(30,000\) epochs, show that \(\gamma\) scales the rate of amplitude decay, while \(\eta\) governs frequency-dependent phase speeds.
Abstract:Modeling the nonlinear acoustic wave propagation in biological tissues needs accuracy, especially in ultrasound imaging and therapeutic applications. The 1D Westervelt equation, \( p_{xx} - p_{tt} + \gamma\, p_{ttt} + \eta\,(p^{2})_{tt}=0, \) could be used to describe a finite-amplitude ultrasound propagation in these tissues. This equation accounts for nonlinearity, \(\eta\), and attenuation, \(\gamma\). To solve this equation, we observe that traditional solvers may struggle to do so correctly. This is due to the difficulties arising from the nonlinear nature of the equation. We solve the equation by defining three synthetic cases - linear (\(\eta=0.001\), \(\gamma=10^{-6}\)), shock-dominated (\(\eta=0.02\), \(\gamma=10^{-6}\)), and dissipation-dominated (\(\eta=0.01\), \(\gamma=5\times10^{-4}\)) - and using Physics-Informed Neural Networks (PINNs) to showcase the solution. A sinusoidal activation function is used to capture the oscillatory behavior of the problem. The results of our study, with PDE residual converging to at least \(10^{-6}\), boundary condition loss to \(10^{-5}\), and the PINN loss converging to \(10^{-5}\) for each case at \(30,000\) epochs, show that \(\gamma\) scales the rate of amplitude decay, while \(\eta\) governs frequency-dependent phase speeds.
Analysis of PDEs (math.AP); Machine Learning (cs.LG); Neural and Evolutionary Computing (cs.NE)
arXiv:2611.05130 [math.AP]
(or arXiv:2611.05130v1 [math.AP] for this version)
https://doi.org/10.48550/arXiv.2611.05130 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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Full text under the publisher's open licence · 276 words