Neural Operators for Immersed-Boundary Soft Swimmers Locomotion

Technical Summary: Neural Operators for Immersed-Boundary Soft Swimmers Locomotion

Problem Statement
High-fidelity immersed-boundary (IB) simulations are essential for analyzing bio-inspired soft underwater swimmers, as they resolve the complex coupling between deforming bodies, nonlinear wake dynamics, and fluid forces. However, the computational cost of repeated fluid–structure interaction (FSI) calculations prohibits their use in engineering design, parameter sweeps, and control applications. While neural operators have shown promise on canonical mathematical benchmarks, their efficacy in high-fidelity engineering workflows involving moving geometries, coupled multiphysics, and volumetric data remains an open question. This study addresses the need for efficient surrogate models that can predict distributed hydrodynamic fields (velocity, vorticity, and pressure) for moving-boundary swimmer flows without the expense of repeated full-physics simulations.

Methodology
The authors develop Fourier Neural Operator (FNO) surrogates to predict the temporal evolution of Eulerian fluid fields generated by planar (2D) and volumetric (3D) eel swimmers. The models are trained on data exported from adaptive IB simulations using the IBAMR framework.

• Data Generation: High-fidelity simulations were performed over a range of Reynolds numbers ($Re$).
  • Planar Case: 50 trajectories on a $256 \times 256$ grid. Training data ($Re \le 7000$) was used to test five held-out trajectories at higher $Re$ (7300–8500), representing high-side extrapolation.
  • Volumetric Case: 50 trajectories on a $64^3$ grid. Training data included $Re$ up to 8500, with five held-out trajectories ($Re \in \{750, 1100, 2200, 4200, 6200\}$) representing interpolation within the training range.
• Model Architecture:
  • 2D Model: A single joint network predicts four outputs simultaneously: velocity components ($u_x, u_y$), scalar vorticity ($\Omega$), and pressure ($P$). Inputs include the previous time step's fields, a binary occupancy mask, Reynolds number, and time step.
  • 3D Model: A suite of three separate, target-specific networks. One network predicts 3D velocity, a second predicts 3D vorticity, and a third predicts pressure. They share a common 15-channel input vector (including flow fields, masks, $Re$, time step, and phase information).
• Training Protocol: Models were trained using Adam optimization with cosine annealing. Loss functions were defined as the mean per-sample joint relative $L_2$ norm of the physical outputs. Evaluation utilized a "teacher-forced" approach, where predictions at time $t_{n+1}$ were conditioned strictly on the reference field at $t_n$.

Key Contributions

1. Formulation: The paper presents a neural-operator surrogate formulation for immersed-boundary swimmer flows in both two and three dimensions, incorporating moving-geometry and flow-parameter conditioning.
2. Quantitative Assessment: It provides a field-resolved, quantitative evaluation of velocity, vorticity, and pressure over held-out Reynolds-number trajectories. The assessment includes full-domain, fluid-region, component-wise, and physical-consistency metrics.
3. Demonstration of Feasibility: The study demonstrates the capability of neural operators to reproduce Eulerian fields for genuinely volumetric, high-fidelity swimmer simulations, including parameter-dependent behavior and slice-resolved 3D reconstruction.

Results

• Planar (2D) Performance: The joint model achieved a full-domain global relative $L_2$ error of 3.51% on the held-out high-Reynolds-number trajectories. The error increased monotonically with $Re$ (from 3.37% at $Re=7300$ to 3.78% at $Re=8500$).
• Volumetric (3D) Performance: The target-specific models achieved the following full-domain global relative $L_2$ errors on held-out within-range trajectories:
  • Velocity: 3.44%
  • Vorticity: 5.59%
  • Pressure: 19.2%
• Physical Consistency: While field-wise errors were low, the data-only training objective did not enforce incompressibility or the curl relationship between velocity and vorticity. Consequently, the predictions exhibited larger divergence and vorticity inconsistency compared to the target fields.
• Pressure Accuracy: Pressure prediction remained the most challenging, with a global error of 19.2% in the 3D case. Analysis indicated that this error was not primarily due to a uniform gauge offset (spatial mean removal did not significantly reduce the error) but likely stemmed from near-body gradients and regime-dependent behaviors.

Significance and Claims
The paper positions immersed-boundary swimmer locomotion as a substantive engineering application for neural operators, moving beyond canonical benchmarks to complex, moving-boundary FSI problems. The authors claim that their results demonstrate the feasibility of field-resolved neural surrogates for these flows, offering a pathway to reduce the cost of repeated model queries in design and control.

However, the authors maintain a modest stance regarding the current state of the technology. They explicitly identify pressure accuracy and physical consistency (differential relationships) as priorities for further development. The study does not claim to have solved the generalization problem for unseen swimmer geometries, gaits, or materials, nor has the surrogate been validated against experimental measurements. The work serves as a concrete baseline (frozen models and reproducible evaluation artifacts) for future research into physics-aware losses, cross-field consistency, and stable temporal prediction in moving-boundary fluid-structure interaction.