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Sensitivity-Guided Parameter Estimation of Bubbly Liquid via a Nonlinear Wave Equation and Physics-InformedNeuralNetworks

Takeru Hasegawa, Naoto Mitsume, Shota Deguchi, Mitsuteru Asai, Tetsuya Kanagawa

arXiv:2610.32668v1physics.flu-dyncs.NAmath.NA

TL;DR

Estimating how physical parameters shape nonlinear ultrasound propagation in bubbly liquids is difficult, and extensive numerical exploration can be costly. This study combines a KdVB equation with a PINN and assesses sensitivity before inverse estimation. Five parameters converge near their true values, while the polytropic exponent only partially converges and surface tension cannot be estimated.

  • Problem

    The effects of medium properties and bubble initial states on ultrasound propagation are difficult to understand systematically, and exploring them numerically can be computationally costly.

    Source: §1 · p. 3

  • Method

    The framework combines a bubbly-liquid KdVB equation with a PINN and assesses coefficient- and solution-level sensitivity before single-parameter inverse estimation.

    Source: §1 · p. 3

  • Results

    Liquid viscosity, bubble initial radius, liquid density, sound speed, and liquid pressure converge to their true values, while the polytropic exponent does not converge sufficiently close and surface tension cannot be estimated.

    Source: §4.3 · p. 7, §5 · p. 8

  • Takeaways & Limitations

    The observed single-parameter estimation outcomes broadly follow the sensitivity analysis, which assesses estimation difficulty in advance.

    Source: §1 · p. 3, §5 · p. 8

    Only single-parameter estimation was confirmed; simultaneous estimation, limited or noisy observations, and application to experimental data remain future work.

    Source: §5 · pp. 8–9

Abstract

from arXiv · show

Interactions between ultrasound and microbubbles underlie a range of biomedical and industrial applications, yet predicting how the physical parameters of the medium and bubbles govern nonlinear wave propagation remains difficult. This study proposes a parameter estimation framework that combines a Korteweg-de Vries-Burgers (KdVB) equation, describing weakly nonlinear wave propagation in a bubbly liquid, with a physics-informed neural network (PINN). The contributions are threefold. First, the PINN reproduces the reference solution of the governing equation in the forward problem, with a near-zero prediction error at the initial time that confirms the initial condition is enforced exactly as intended. Second, prior to inverse estimation, a sensitivity analysis is introduced that quantifies, at both the coefficient and solution levels, how each of eight physical parameters influences the KdVB equation, allowing the expected difficulty of estimating each parameter to be assessed in advance. Third, based on this assessment, single-parameter inverse estimation is performed for seven of the eight parameters using the PINN. Liquid viscosity, bubble initial radius, liquid density, sound speed, and liquid pressure are estimated with errors of 0.8-2.1%, the polytropic exponent converges only partially, and the surface tension coefficient cannot be estimated at all, consistent with the sensitivity analysis. These results establish a general framework for identifying and estimating the physical parameters governing nonlinear wave propagation in bubbly liquids from observed waveform data.

1. Introduction

The study combines a bubbly-liquid KdVB model with a PINN to estimate physical parameters from waveforms. It assesses parameter sensitivity before inverse estimation, addressing the difficulty of understanding parameter effects and the computational cost of extensive simulation.

1. Introduction

  • Empirical observation alone does not systematically reveal how medium properties and initial bubble states affect propagation, while numerical exploration can be computationally costly.Estimating governing parameters directly from observed waves is proposed as a way to support systematic understanding and safe irradiation design.Source: §1 · pp. 2–3
  • The KdVB model expresses nonlinear, dissipative, and dispersive coefficients explicitly as functions of physical parameters, providing a basis for waveform-based parameter estimation.The study situates this model in a lineage of reduced-order nonlinear wave equations for media containing bubbles.Source: §1 · p. 3
  • Prior PINN work addressed weakly nonlinear wave equations and other inverse problems, but the authors identify no prior pre-estimation sensitivity assessment for PINNs applied to this class of wave equation.The cited KdV study used a dispersive-only equation, whereas the present KdVB model also includes dissipation.Source: §1 · p. 3
  • The study combines a bubbly-liquid KdVB equation with a PINN, first validating forward prediction, then assessing parameter sensitivity and performing single-parameter inverse estimation.The sensitivity analysis evaluates both equation coefficients and the solution to estimate parameter-estimation difficulty in advance.Source: §1 · p. 3

2. Governing equation

The study uses the KdVB equation to describe nonlinear wave propagation in a bubbly liquid. Its nonlinear, dissipative, and dispersive coefficients depend on eight physical parameters.

2. Governing equation

  • The KdVB equation is used to model nonlinear wave propagation in a bubbly liquid.The study draws on the model of Yu et al. (2025).Source: §2 · p. 4
  • The nonlinear, dissipative, and dispersive coefficients depend on eight physical parameters and are independent of space and time.The parameters are bubble initial radius and number density, polytropic exponent, liquid density, sound speed, surface tension, liquid viscosity, and liquid pressure.Source: §2 · p. 4

3. Methodology

The PINN is designed to satisfy the initial and periodic boundary conditions exactly rather than imposing them as soft loss penalties. Its loss therefore omits separate initial-condition and boundary-condition penalties.

3. Methodology

  • In a standard PINN, initial and boundary conditions enter the loss as penalties, so minimizing the loss may still leave residual violations.The periodicity penalty covers the predicted solution and its spatial derivatives at the domain boundaries.Source: §3 · p. 4
  • Fourier feature embedding makes the network output exactly periodic in space.The embedding uses sine and cosine features at integer multiples of πx.Source: §3 · p. 4
  • The network output is formed by multiplying its learned component by time and adding the initial-condition function, enforcing that condition exactly.As a result, the initial-condition and periodic-boundary penalty terms are unnecessary.Source: §3 · pp. 4–5

4. Results and discussion

The forward PINN reproduces the reference wave, with error increasing over time but an extremely small initial-time error. Sensitivity patterns broadly anticipate inverse-estimation accuracy: five parameters converge near their true values, while the polytropic exponent only partially converges and surface tension does not.

4.1. Forward problem

  • The forward experiment trains the KdVB-informed PINN for 100,000 epochs across five random seeds, with loss becoming nearly flat after 50,000 epochs.The network uses 3,000 collocation points and 600 observational data points, resampled at each epoch.Source: §4 · p. 5, §4.1 · p. 6
  • The PINN reproduces the reference solution's overall trend, but absolute error increases over time and clear deviations appear at t = 0.75 and t = 1.0.At t = 0, relative L2 error and its standard deviation are extremely small, consistent with exact initial-condition enforcement.Source: §4.1 · p. 6

4.2. Sensitivity analysis

  • The sensitivity analysis numerically differentiates both coefficients and solutions using central differences with a step of 10^-4 relative to normalized parameters.The same scheme is used for both quantities for methodological consistency.Source: §4.2 · p. 6
  • Most parameters affect the dissipative and dispersive coefficients, while the nonlinear coefficient depends little on parameters other than the polytropic exponent.Bubble number density has no coefficient sensitivity because none of the governing-equation coefficients depend on it.Source: §4.2 · p. 7
  • Solution sensitivity tends to track estimation difficulty: viscosity and initial bubble radius are expected to estimate well, while surface tension is expected to be harder.The equation and its initial and boundary conditions are independent of bubble number density, making its estimation fundamentally impossible in principle.Source: §4.2 · p. 7

4.3. Inverse problem

  • Each inverse run estimates one parameter while fixing the others at their true values, using initial estimates of either 0.5 or 2.0 times the true value.Training uses 100,000 epochs and five random seeds; an exponential parameterization keeps the estimated parameter positive.Source: §4.3 · p. 7
  • Liquid viscosity, bubble initial radius, liquid density, sound speed, and liquid pressure converge to their true values; the polytropic exponent remains insufficiently close and surface tension estimation fails.The five successful estimates are within approximately 2% after 100,000 epochs; the error ordering is broadly consistent with solution sensitivity, except for the polytropic exponent.Source: §4.3 · pp. 7–8, Table 2 · p. 8

5. Summary

The study combines a KdVB equation with a PINN and uses sensitivity analysis to guide single-parameter estimation. Estimation accuracy followed the predicted sensitivity, while the study identifies limits to the demonstrated scope.

5. Summary

  • The framework combines the KdVB equation for nonlinear ultrasound propagation in bubbly liquid with a PINN and assesses parameter sensitivity at coefficient and solution levels before inverse estimation.The analysis was used to assess expected estimation difficulty in advance.Source: §5 · p. 8
  • Errors were approximately 0.8–2.1% for liquid viscosity, bubble initial radius, liquid density, sound speed, and liquid pressure; the exponent converged partially and surface tension was not estimated.Bubble number density is analytically shown to be impossible to estimate from the governing equation because it does not affect the solution.Source: §5 · p. 8
  • The forward problem reproduced the reference solution, with near-zero initial-time error confirming that the initial condition was enforced as intended.The study reports this alongside the sensitivity and inverse-estimation findings.Source: §5 · p. 8
  • The demonstrated inverse estimation is single-parameter; simultaneous estimation and estimation from limited, noisy observations remain needed for broader applicability.The eight parameters enter through only three dimensionless groups, so different parameter combinations may yield similar solutions and complicate separate estimation.Source: §5 · pp. 8–9

Appendix A. Explicit form of the coefficients Π1, Π2, and Π3 and the nondimensional phase

Appendix A presents explicit coefficient expressions and defines quantities used in the nondimensional formulation, including the phase velocity and bubble-related terms.

Appendix A. Explicit form of the coefficients Π1, Π2, and Π3 and the nondimensional phase

  • The appendix introduces explicit expressions for the nonlinear, dissipative, and dispersive coefficients.The supplied passage begins the coefficient expressions but does not give enough complete text to summarize their forms.Source: Appendix A · p. 9
  • The passage defines void fraction and ω as a nondimensional parameter related to wave frequency, and identifies the undisturbed gas pressure and bubbles’ natural angular frequency.The gas pressure is that inside a bubble without disturbance.Source: Appendix A · p. 9
  • U* denotes the typical phase velocity of the wave.Source: Appendix A · p. 10

Appendix B. Independence of the governing equation on the bubble number density

Appendix B shows that bubble number density does not affect the KdVB coefficients or solution, making its estimation from the solution impossible in principle.

Appendix B. Independence of the governing equation on the bubble number density

  • For Π1, the appendix differentiates the coefficient with respect to void fraction and shows the terms cancel, leaving the coefficient independent of bubble number density.The derivation proceeds through the coefficients’ dependence on void fraction.Source: Appendix B · p. 10
  • The appendix also establishes that Π2 and Π3 are independent of void fraction, and therefore of bubble number density.For Π2, the phase velocity in the prefactor is itself independent of void fraction.Source: Appendix B · p. 10
  • The governing equation and its initial and boundary conditions are independent of bubble number density, so the solution cannot depend on it and it is impossible to estimate from solution observations.The passage relates this to null sensitivity in both coefficients and solution.Source: Appendix B · p. 10

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