VesNet: Neural network accelerated solver for simulating Stokesian vesicle suspensions

Technical Summary: VesNet: Neural network accelerated solver for simulating Stokesian vesicle suspensions

Problem Statement
Numerically simulating suspensions of deformable particles, specifically vesicles, in the Stokes flow regime is computationally prohibitive. These systems are characterized by strongly nonlinear fluid–structure interactions, evolving interfaces, and multiscale hydrodynamic phenomena involving long-range many-body effects and short-range lubrication forces. Traditional high-fidelity methods, such as the Boundary Integral Equation Method (BIEM), require solving large linear systems at every time step and handling expensive near-singular integrals, making them difficult to scale for dense suspensions or long-time horizons. While machine learning (ML) has been applied to particulate flows, most existing approaches focus on rigid particles or act as purely surrogate models that lack the physical fidelity required for deformable bodies with complex hydrodynamic coupling.

Methodology: VesNet Framework
The authors introduce VesNet, a hybrid solver that integrates neural networks (NNs) with conventional algorithmic modules to accelerate vesicle simulations in two dimensions. Rather than replacing the entire physics solver, VesNet uses NNs to approximate specific, computationally expensive integro-differential operators while retaining standard algorithms for boundary reparameterization and far-field interactions.

• Governing Equations and Operator Splitting: The solver addresses the coupled fluid-solid interaction equations for inextensible membranes resisting bending. The time-stepping scheme employs operator splitting to decouple the advection (driven by background flow and inter-vesicle interactions) and relaxation (driven by membrane elasticity) steps.
• Neural Operator Approximation: The core innovation lies in approximating the nonlinear operators using the Iterated V-shape Net (IVNet), a multilevel convolutional architecture similar to U-Net.
  • Self-Interactions: IVNet approximates the flow map for the relaxation step and the tension calculation, which depend solely on a single vesicle's configuration.
  • Far-Field and Advection: To ensure generalization to arbitrary flow conditions, the velocity field is decomposed into a truncated Fourier series. Separate subnetworks are trained to map vesicle configurations to the response of each Fourier mode, allowing the solver to handle unseen background flows and particle counts without retraining.
  • Near-Singular Integration: A novel NN-based scheme replaces traditional singular quadrature for near-field interactions. The method constructs five concentric layers around a vesicle membrane (including the membrane itself and layers at $\pm h/2$ and $\pm h$). NNs approximate the single-layer integral on these layers, and Radial Basis Function (RBF) interpolation is used to recover velocities at arbitrary points in the near-field. This avoids the need for expensive direct evaluation of singular integrals during simulation.
• Physical Invariances and Training: The networks are trained on short-time horizon simulations of single vesicles in various flow conditions (shear, Poiseuille, rotating cylinders). To ensure generalizability, input data is standardized for translation, rotation, and scaling invariance. The networks are trained to minimize relative error in both the solution and its derivatives to enhance smoothness.
• Implementation: The entire pipeline, including solver steps and network inference, is implemented on GPUs using PyTorch. The code utilizes batching, vectorized tensor operations, and batched linear algebra (e.g., Cholesky decomposition) to maximize parallelism.

Key Contributions

1. Hybrid Solver Architecture: VesNet combines the accuracy of BIEM with the speed of ML by using NNs specifically for the most expensive operators (near-singular integrals and self-interactions) while retaining exact algorithms for far-field interactions and geometric constraints.
2. IVNet for Stokes Flow: The application of Iterated V-shape Nets to approximate nonlinear operators in Stokesian particulate flows, demonstrating their efficacy in capturing multiscale nature and convolutional structures inherent in the fundamental solution.
3. Neural Near-Singular Integration: A new strategy for handling near-field interactions that eliminates the need for singular quadrature rules during the simulation, replacing them with NN approximations and RBF interpolation.
4. Generalization via Fourier Decomposition: A method to decouple the network from specific flow conditions by training on Fourier modes of the velocity field, enabling the solver to handle arbitrary background flows and suspension densities without retraining.
5. GPU-Accelerated Performance: A fully GPU-accelerated implementation that achieves significant speedups over both CPU and GPU versions of state-of-the-art BIEM solvers.

Results and Performance
The authors evaluate VesNet against ground-truth BIEM simulations across several scenarios:

• Relaxation: VesNet accurately recovers the equilibrium shape and bending energy of a deformed vesicle in a stationary fluid, even for configurations not present in the training dataset.
• Poiseuille Flow: The solver correctly predicts the lateral migration and equilibrium configurations of vesicles under varying confinement and capillary numbers. VesNet with $N=32$ discretization points achieves accuracy comparable to or better than BIEM with the same resolution, and approaches the accuracy of BIEM with $N=128$.
• Shear Flow (Vesicle Pairs): In simulations of two interacting vesicles, the proposed near-singular integration scheme is shown to be essential for stability. Without it, collisions occur erroneously; with the NN correction, VesNet accurately captures the separation distance and avoids collisions, matching high-resolution BIEM results.
• Dense Suspensions: Simulations of 2,000 vesicles in Taylor-Green and Poiseuille flows demonstrate that VesNet reproduces key observables (e.g., cell-free layer formation, migration statistics) with high fidelity.
• Speedup: VesNet delivers a 100$\times$ speedup over a multithreaded MATLAB CPU implementation of a fully resolved BIEM solver and a 5$\times$ speedup over a GPU implementation of the same BIEM solver. A single time step for 2,000 vesicles takes 1.4 seconds on an NVIDIA GH200.

Significance and Limitations
The paper claims that VesNet provides a robust tool for parameter exploration, optimization, and uncertainty quantification in vesicle suspensions, scenarios where traditional simulations are often too slow. By achieving BIEM-level accuracy at reduced spatial and temporal resolutions, VesNet enables extensive simulations on relatively modest computational resources.

The authors explicitly note current limitations:

• The framework is restricted to 2D simulations.
• It does not currently accommodate viscosity contrasts between the interior and exterior fluids of the vesicle.
• It cannot guarantee collision-free interactions in all regimes, a challenge that remains difficult even for standard BIEM.
• The solver assumes the fluid inside the vesicle is identical to the suspending fluid.

The work establishes a foundational framework for extending these hybrid ML-physics approaches to 3D systems and more complex biological and industrial applications.