Operator learning for predicting multiscale bubble growth dynamics

Chensen Lin; Zhen Li; Lu Lu; Shengze Cai; Martin Maxey; George Em Karniadakis

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Operator learning for predicting multiscale bubble growth dynamics

Chensen Lin, Zhen Li, Lu Lu, Shengze Cai, Martin Maxey, George Em Karniadakis

arXiv:2012.12816v1Dec 23, 2020physics.comp-ph

TL;DR

Multiscale bubble-growth prediction requires connecting deterministic macroscale and stochastic microscale descriptions, a task not systematically investigated with DNNs. The paper develops and tests DeepONet on R-P and DPD bubble-growth models, finding that it can unify the two regimes while greatly reducing computational time for noisy data.

Problem
Deep neural networks have not been systematically evaluated for multiscale problems coupling heterogeneous physics across deterministic macroscale and stochastic microscale regimes.
Method
The study tests DeepONet as a surrogate for macroscale R-P bubble dynamics and microscale stochastic DPD nucleation and cavitation.
Results
DeepONet demonstrated effectiveness for both the R-P and DPD models, with noisy-data computation reduced from 48 CPU hours to a fraction of a second, a speed-up over 200,000.
Takeaways & Limitations
The findings support using DeepONets to unify macroscale and microscale models of multirate bubble growth.

Abstract

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Simulating and predicting multiscale problems that couple multiple physics and dynamics across many orders of spatiotemporal scales is a great challenge that has not been investigated systematically by deep neural networks (DNNs). Herein, we develop a framework based on operator regression, the so-called deep operator network (DeepONet), with the long term objective to simplify multiscale modeling by avoiding the fragile and time-consuming "hand-shaking" interface algorithms for stitching together heterogeneous descriptions of multiscale phenomena. To this end, as a first step, we investigate if a DeepONet can learn the dynamics of different scale regimes, one at the deterministic macroscale and the other at the stochastic microscale regime with inherent thermal fluctuations. Specifically, we test the effectiveness and accuracy of DeepONet in predicting multirate bubble growth dynamics, which is described by a Rayleigh-Plesset (R-P) equation at the macroscale and modeled as a stochastic nucleation and cavitation process at the microscale by dissipative particle dynamics (DPD). Taken together, our findings demonstrate that DeepONets can be employed to unify the macroscale and microscale models of the multirate bubble growth problem, hence providing new insight into the role of operator regression via DNNs in tackling realistic multiscale problems and in simplifying modeling with heterogeneous descriptions.

I. INTRODUCTION

The paper examines whether DeepONet can support multiscale bubble-growth modeling across deterministic macroscale and stochastic microscale descriptions. It targets the difficulty of coupling heterogeneous models whose governing equations and valid scale ranges differ.

Deep neural networks have mainly been applied to single-scale physical problems, while truly multiscale problems remain challenging.
Different mathematical formulations across scales make heterogeneous physical models difficult to integrate and couple.
DeepONet is proposed as a possible unified framework because its learned physical models lack explicit governing equations.
The study tests multiscale bubble growth described by the Rayleigh-Plesset equation at the continuum scale and stochastic nucleation and cavitation at the microscale.
The macroscale model uses the R-P ordinary differential equation under continuum and spherical-bubble assumptions, while microscale dynamics are modeled with stochastic DPD equations.
The paper’s goal is to test DeepONet as a surrogate model for both macroscale and microscale physics.

II. LEARNING OPERATORS VIA DEEPONET

DeepONet learns an operator mapping discretized ambient-pressure trajectories to continuous bubble-radius predictions. Its branch and trunk networks encode the input function and output coordinate, respectively.

DeepONet approximates nonlinear continuous operators by combining a branch network for input functions with a trunk network for output coordinates.
In the bubble-dynamics application, the input function is ambient liquid-pressure change ΔP and the output function is bubble radius R(t).
The branch network receives pressure-change values at fixed sensor times, while the trunk network receives time t.
The branch and trunk outputs are merged to predict bubble radius as an operator evaluation R(t) = G(Δp)(t).
Training minimizes mean squared error between the R-P bubble radius and the network prediction for pressure-trajectory and time inputs.
DeepONet requires fixed input-function evaluation locations but can estimate the output radius continuously at arbitrary times.

A. Rayleigh–Plesset model

The Rayleigh–Plesset model describes spherical bubble-radius dynamics in an infinite liquid under continuum assumptions, with pressure forcing producing expansion, oscillation, damping, and equilibration.

Model formulation: The R-P equation models the radius R(t) of a spherical gas-vapor bubble in an infinite liquid as far-field pressure varies over time.It is a continuum-level model derived for a spherically symmetric bubble in an incompressible fluid.
Model formulation: The model balances liquid inertia, viscous damping, surface tension, and pressure at the bubble surface and in the far field.The liquid is assumed incompressible, with constant density and kinematic viscosity, and at rest far from the bubble.
Bubble response: A sharp far-field pressure drop induces rapid bubble expansion followed by damped oscillations and gradual adjustment to a new equilibrium size.Without viscous dissipation, the oscillations would continue indefinitely without attenuation.
Input construction: The resulting pressure and radius trajectories represent characteristic bubble responses as the bubble passes through a low-pressure region.The setup uses representative water–air parameters and serves as the starting point for operator learning on bubble dynamics.

B. Convergence tests

The convergence tests examine how DeepONet architecture and training-data organization affect error, then assess prediction quality on in-distribution trajectories and an out-of-distribution pressure signal.

Architecture convergence: Network width has an optimum: increasing width can reduce error initially, but further increases can raise the error.The width study uses depth 2 networks to examine this behavior.
Architecture convergence: Increasing depth lowers training error, while test loss reaches its minimum at about 6 hidden layers for width 100.This indicates that training and test error respond differently to increasing network depth.
Training-data convergence: The number of training trajectories is usually more important than the density of data points per trajectory when the total number of data points is fixed.Increasing the total amount of training data generally reduces error, but trajectory diversity has the stronger effect under a fixed data-point budget.
Prediction performance: With a network width of 200, depth 3, and 1×10^5 data points from 5000 trajectories, training and testing losses converge to small values.The model is trained with Adam using a learning rate of 5 × 10^-4 and ReLU activations.
Prediction performance: The trained DeepONet predicts bubble radius at arbitrary times from input functions and accurately captures oscillations, decay, and subsequent smooth changes.Representative predicted trajectories closely follow the ground truth, and the model is also evaluated on a pressure input not drawn from the Gaussian random-field distribution.

C. Comparison of DeepONet and LSTM

DeepONet and LSTM both handle the bubble-growth forecasting problem with ample training data, but DeepONet is more flexible when trajectory samples are sparse or irregularly located.

Both DeepONet and LSTM handle the problem well when training data are ample and hyperparameters are tuned.
DeepONet accepts arbitrary numbers and locations of output-function evaluation points, unlike LSTM’s aligned-grid requirement.The comparison uses 20, 80, and 200 points per trajectory, with 5000 training trajectories in each case.
DeepONet shows a great advantage when training data are very sparse.
DeepONet predicts accurately from arbitrary locations, including sparse samples, and its trajectory prediction does not deviate from the ground truth in the 20-point case.
With ample training data, DeepONet can produce smooth and accurate results even when only a single data point is collected per trajectory.
DeepONet therefore offers advantages in accuracy and versatility when measurement intervals are unknown or dense data are difficult to acquire.

D. Predictions for random initial bubble size

DeepONet predicts bubble-radius dynamics across randomly varied initial bubble sizes, capturing the resulting changes in initial radius, oscillation frequency, magnitude, and decay.

The training dataset contains 30,000 trajectory pairs with random initial bubble sizes ranging from 1 × 10^-5 m to 2 × 10^-5 m.
Initial bubble size strongly affects both the initial solution value and the oscillation frequency.
DeepONet predictions agree very well with ground truth for random initial bubbles.
The model accurately predicts different oscillation magnitudes, frequencies, and decay behaviors in test trajectories.

E. Extrapolation: Predicting inputs outside the function space of the training data

DeepONet can extrapolate to pressure fields outside the training function space, with performance depending on how the input space is represented and improving after sparse-data fine-tuning.

Pretrained DeepONet predictions can have relatively high L2 errors on unseen inputs before adaptation.
For unseen correlation lengths l ranging from 0.07 to 0.9, extrapolation error rises moderately with increasing l but can increase sharply as l decreases.
Training on l = 0.07, 0.1, and 0.15 reduces errors at small l but increases errors at large l, balancing performance across the range.
A better representation of the input space leads to higher extrapolation accuracy.
Fine-tuning with sparse measurements selected at pressure-profile extrema improves agreement with ground truth.Only the last trunk-net layer is tunable during extrapolation fine-tuning.
15.3% falls to 9.7% with 3 new data or 6.2% with 6 new data in the first case, while 19.5% falls to 14.3% or 11.1% in the second.These are relative L2 errors before and after sparse-data fine-tuning.
Transfer learning with sparse data produces better solutions than the original pretrained networks and takes only a few seconds.

IV. MICROSCALE (STOCHASTIC) REGIME

The microscale regime uses DPD/MDPD particle simulations to model stochastic bubble growth under time-varying pressure, then trains DeepONet to infer bubble-radius dynamics from pressure inputs. The learned model captures the mean behavior of noisy trajectories and evaluates predictions vastly faster than direct simulation.

Microscale model: DPD models stochastic microscale bubble dynamics with dissipative, conservative, and random particle forces, while MDPD adds attractive interactions for the liquid phase.The model includes fluctuation-dissipation constraints and uses separate gas and liquid interaction descriptions.
Simulation setup: A moving top wall applies preset time-varying pressure, and gas-phase volume is estimated with Voronoi tessellation to obtain the effective bubble radius.The simulation records gas volume while the external pressure evolves from an initial equilibrium state.
Training data: 400 independent DPD simulations generate training pairs, with pressure trajectories as inputs and bubble-radius trajectories as outputs.DeepONet uses 50 evenly spaced input sensors and 25 randomly spaced output points.
Prediction performance: The microscale data contain high-frequency fluctuations, but DeepONet predictions form smooth trajectories that capture the data mean accurately.Parity results place predictions near the diagonal, with discrepancies attributed mainly to noisy raw trajectories versus smooth predictions.
Computational gain: DeepONet evaluates the microscale response in a fraction of a second versus 3 hours for each 16-core DPD simulation, yielding a speed-up exceeding 200,000.The learned operator infers bubble growth from complex particle simulations after training.

V. SUMMARY

The study demonstrates DeepONet across deterministic macroscale and stochastic microscale bubble-growth models, emphasizing flexible sampling, transfer learning, and substantial computational savings. These results support a future unified model spanning the two regimes, while leaving seamless cross-scale prediction for future work.

Model scope: DeepONet is evaluated on both the Rayleigh–Plesset macroscale model and the DPD microscale model for time-varying bubble growth.The study uses Gaussian random fields to generate pressure inputs for the dynamical systems.
Data flexibility: Compared with LSTM, DeepONet offers flexibility in the density and location of training data.This is useful when the sufficient data density is unknown or data acquisition is restricted.
Generalization: When the pressure correlation length falls outside the training range, predictions are not very accurate, but transfer learning and trunk-net fine-tuning improve the situation with few new data points.The paper explicitly tests inputs with unseen correlation lengths.
Microscale result: At the microscale, DeepONet learns the mean component of noisy raw data without extra processing and reduces computation from 48 CPU-hours to a fraction of a second.The reported speed-up exceeds 200,000.
Implication: The results establish a foundation for a unified neural-network model that could predict bubble-growth dynamics from macroscale to microscale seamlessly.The paper states that such seamless cross-scale results remain under investigation.