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A Neural-Network Approach for the Stability of Loosely-Coupled FSI Simulations in Hemodynamics

Francesca Renzi, Stefano Pagani, Paola F. Antonietti, Christian Vergara

arXiv:2610.10943v1math.NAcs.NA

TL;DR

High added-mass can destabilize loosely coupled FSI schemes, while choosing the LC-DN-α scheme’s stabilizing parameter for a particular problem remains difficult. The paper trains a feed-forward neural network on baseline FSI problems to estimate α, and reports generalization across tested scenarios with robust estimates under moderate geometry and loading changes.

  • Problem

    Loosely coupled DN schemes can be unstable in high-added-mass regimes, and the optimal α for LC-DN-α depends on FSI problem characteristics without a practical selection criterion.

    Source: §1 · p. 2, §2.2 · p. 5

  • Method

    A feed-forward neural network is trained on baseline 2D FSI problems in large added-mass regimes to estimate α from problem descriptors.

    Source: §1 · p. 3

  • Results

    The model generalizes across considered test cases and provides robust α estimates under moderate changes in loading conditions and computational domain.

    Source: §6 · p. 20

  • Takeaways & Limitations

    Reliable estimates under common geometry and inlet-condition variations are an encouraging step toward reducing computational demand in patient-specific hemodynamics.

    Source: §6 · p. 20

    The methodology is developed and validated for simplified 2D problems with Stokes flows and linear-elastic structures; extending it to Navier–Stokes flow and nonlinear structural models may require additional input features.

    Source: §6 · p. 20

Abstract

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The development of accurate and efficient computational methods for fluid-structure interaction (FSI) problems remains an active research area due to its broad engineering and biomedical applications. Partitioned loosely coupled (LC) schemes are attractive for large-scale simulations because of their modularity and low computational cost. However, their applicability is limited by stability issues in high added-mass regimes, where fluid and structure densities are comparable, as in hemodynamics. In our recent work, we proposed an LC algorithm based on Dirichlet-Neumann interface coupling, the LC-DN-{\alpha} scheme, which improves the stability of the coupling through a stabilization parameter {\alpha}. Although theoretical stability has been demonstrated under suitable constraints on {\alpha}, its optimal value depends on the physical and numerical characteristics of the FSI problem and is difficult to determine in practice. In this work, we propose a data-driven strategy to automatically estimate the optimal value of {\alpha}. We develop a feed-forward neural network to predict {\alpha} from the descriptors of the FSI problem. The model is trained on a dataset of baseline FSI problems in large added-mass regimes and evaluated on more realistic configurations to assess robustness and generalization. Numerical results show that the proposed model generalizes across the considered FSI scenarios and provides robust estimates of {\alpha} under moderate variations in the problem setup, including changes in geometry and loading conditions, that do not completely alter the interaction mechanics represented in the training set. The methodology is intentionally developed and validated on simplified FSI problems, providing a proof of concept for data-driven strategies to improve the efficiency of partitioned FSI simulations and supporting their extension to more complex and realistic scenarios.

1 Introduction

Partitioned loosely coupled FSI methods are attractive for their modularity and low computational cost, but instability in high added-mass regimes limits their use in hemodynamics. This work proposes a neural network to estimate the stabilization parameter for an LC-DN-α scheme, addressing the practical difficulty of choosing it for a given problem.

1 Introduction

  • FSI discretization is challenging because nonlinear coupling, subsystem nonlinearities, and ill-conditioned linear systems can complicate accurate and efficient simulation.The paper situates its approach within both physics-based and data-driven numerical strategies.Source: §1 · p. 2
  • Partitioned methods solve fluid and structure subproblems separately, simplifying implementation and allowing reuse of standalone solvers and preconditioners.Their modularity can be useful when commercial software packages are used.Source: §1 · p. 2
  • Dirichlet-Neumann coupling imposes interface velocity on the fluid as a Dirichlet condition and interface stresses on the structure as a Neumann condition.Strongly coupled schemes use interface iterations; loosely coupled schemes perform one fluid and one structure solve per time step.Source: §1 · p. 2
  • LC-DN schemes retain partitioned-method modularity and DN coupling simplicity while reducing computational cost, but become unstable when fluid and structure densities are comparable.The added-mass effect arises from strong inertial interaction and is especially pronounced in hemodynamics.Source: §1 · p. 2
  • Choosing the optimal stabilization parameter for realistic LC-DN-α simulations remains an open practical challenge despite proven stability ranges.The useful range depends on the problem’s physical and numerical characteristics.Source: §1 · pp. 2–3
  • Scientific Machine Learning for FSI has chiefly addressed optimization and surrogate modeling, while solver-efficiency approaches have focused almost exclusively on strongly coupled schemes.The potential of SciML for loosely coupled schemes remains largely unexplored.Source: §1 · p. 3
  • A feed-forward neural network estimates the LC-DN-α stabilization parameter from FSI problem descriptors, trained on baseline 2D high-added-mass problems and evaluated on more realistic configurations.The study presents this as a proof of concept for applying Scientific Machine Learning to loosely coupled FSI algorithms.Source: §1 · p. 3

2 Algorithm formulation and stability of the LC-DN-α scheme

The LC-DN-α algorithm performs one fluid and one structure solve per time step, with α modifying the right-hand sides to improve stability under added-mass effects. Its stability bound depends on the problem, so the section motivates a data-driven framework to estimate α.

2.1 Formulation of the scheme

  • The fluid is formulated in an ALE framework and the structure in a Lagrangian framework, with interface velocity continuity and traction equilibrium coupling them.The fluid uses incompressible Navier-Stokes equations and the structure uses linear elasticity equations.Source: §2.1 · p. 3
  • Time discretization uses a uniform finite-difference partition, explicit extrapolation for geometric coupling, semi-implicit Euler linearization of convection, and implicit BDF1 structural dynamics.The time step is denoted by Δt.Source: §2.1 · pp. 3–4
  • At each time step, the LC-DN-α algorithm performs one fluid solve and one structure solve, modifying their right-hand sides using α.The scheme retains the partitioned structure while providing stability properties under significant added-mass.Source: §2.1 · p. 4
  • The scheme solves the fluid with a Dirichlet interface condition and the structure with a Neumann interface condition, following the LC-DN coupling sequence.The formulation considers an incompressible Newtonian fluid and a linear elastic structure.Source: §2.1 · pp. 3–4

2.2 Stability of the LC-DN-α scheme

  • The stability theorem states unconditional absolute stability for α below a critical bound involving structural and fluid properties and the maximum eigenvalue of the added-mass operator.The theorem is presented for a benchmark with a rectangular fluid grid, a one-dimensional membrane, linear incompressible inviscid fluid, and a 1D independent-rings structure model.Source: §2.2 · pp. 4–5
  • When ρsHs < ρfµmax, the critical value α* is below 1, making the classical DN choice α = 1 unstable; sufficiently small α restores unconditional stability.The classical DN scheme is recovered by setting α = 1.Source: §2.2 · p. 5
  • The stability analysis gives no practical criterion for the optimal α, whose performance depends on problem characteristics and may vary further in more complex settings.The text notes that nonlinear effects, complex geometries, and additional physical parameters can affect coupling behavior.Source: §2.2 · p. 5
  • The proposed data-driven framework aims to estimate α by analyzing how physical, numerical, and algebraic problem features relate to its optimal value.The analysis identifies relevant features before developing a model to predict α in the algorithm.Source: §2.2 · p. 5

3 The learning framework

The framework defines an optimal stabilization parameter by comparing the partitioned LC-DN-α solution with a monolithic reference over stable α values. It trains feed-forward neural networks to estimate that parameter from descriptors of each FSI problem instance.

3 The learning framework

  • Each instance descriptor d collects geometric, physical, and discretization data, while Figure 1 depicts the baseline problem’s fluid and structure domains.Examples of descriptors include geometry and structural thickness, densities and viscosity, and mesh and time-step sizes.Source: §3 · p. 5, Figure 1 · p. 7
  • The optimal α is the stable value that minimizes the difference between the LC-DN-α and monolithic solutions in a suitable norm.The monolithic solution serves as the reference; the norm may be the H1 norm.Source: §3 · p. 6
  • Estimating α∗(d) requires multiple LC-DN-α solves, motivating a neural network to approximate the mapping from problem parameters to the optimal value.The network’s estimated stabilization parameter is bα.Source: §3 · p. 6
  • The supervised dataset pairs each problem instance with α∗, using monolithic solutions as references and retaining stable candidate α values for the LC-DN-α solves.For each instance, the optimal value is computed from the retained stable candidates.Source: §3 · p. 6
  • Because descriptor components may differ in influence and redundant features can affect network performance, the study analyzes α∗’s dependence on d and compares three input representations.The representations are subsets of d, informed by problem data and selected descriptors of the discrete FSI system’s algebraic structure.Source: §3 · p. 6

4 Results: problem setup and neural network architecture

The section defines a baseline 2D vessel FSI problem, builds and characterizes a dataset of configurations, and trains feed-forward networks to estimate the stabilization parameter α. The models achieve comparable solution accuracy, while the input analysis identifies nonlinear relationships and correlated descriptors.

4.1 The baseline problem

  • The baseline is a 2D rectangular vessel section with incompressible Stokes flow and a linear elastic wall; training uses reflecting conditions because the absorbing case proved harder to train.The network's effectiveness in the absorbing case is assessed later in Section 5.Source: §4.1 · p. 7

4.2 Generation of the learning dataset

  • The 361-configuration dataset varies mesh size h, time step Δt, Young modulus E, and structural density ρs, with samples selected by optimized Latin Hypercube Sampling.The sampled configurations and their optimal α* values form the learning dataset.Source: §4.2 · p. 8

4.3 Input space characterization

  • E and Δt correlate strongly and positively with α*, while ρs and h show very weak correlations; algebraic descriptors correlate strongly with α* but are also correlated with E and Δt.Source: §4.3 · p. 9, Figure 3 · p. 10
  • The three candidate input sets combine physical and numerical parameters with matrix or Schur-complement descriptors to represent stabilization and coupling effects.The sets are DI = {Δt, E, ρs}, DII = {Δt, ||As||F, ||Af||F}, and DIII = {Δt, E, ρs, ||Σ||F}.Source: §4.3 · pp. 9–10

4.4 Selection and training of the neural network model

  • Three FFNNs share an architecture and training procedure but use different input sets; each has one 10-neuron ReLU hidden layer and a linear output.They were trained for 2500 epochs, and training and test losses converged to stable, close plateaus.Source: §4.4 · pp. 10–11
  • All three models yield comparable LC solution accuracy, although fθ,I has the lowest training/test loss and slightly better error distributions.The overlapping interquartile ranges indicate errors of the same order for most tested configurations, and feature choice has limited influence on final accuracy.Source: §4.4 · p. 11

5 Results: robustness of the neural network

The network’s estimated stabilization parameter performs well on unseen test configurations and under moderate inlet and geometry changes. Performance degrades when an absorbing outlet condition changes the fluid–structure interaction pattern.

5.2 Robustness with respect to changes in the baseline problem

  • 44.4% of simulations remained stable with an absorbing outlet; stable cases had median relative errors of nearly 50% for ux, 30% for p, and 3% for df.The authors attribute the degradation to changed fluid dynamics and recommend new training for the absorbing-outlet case.Source: Figure 5 · p. 13, §5.2 · p. 16

5.1 Robustness with respect to the test set

  • Predicted LC-DN-bα solutions stay close to optimal-α solutions across three unseen scenarios, although scenario (c) shows larger discrepancies from the monolithic solution.The largest monolithic discrepancies are attributed mainly to differences between the loosely coupled and monolithic formulations, rather than to network prediction errors.Source: §5.1 · p. 13, Figure 6 · p. 14

5.2 Robustness with respect to changes in the baseline problem

  • With stenosis, stability was almost completely preserved across geometries, and median relative errors were lower than for the baseline geometry.The authors suggest flow restriction may partially compensate for the added-mass effect; Table 3 reports stable-simulation percentages by stenosis depth and width.Source: §5.2 · pp. 16, 18, Table 3 · p. 18
  • 100% of cases with pin,1, pin,2, and pin,4 and 99.8% with pin,3 remained stable; median errors were below 10% for ux, 16% for p, and 0.3% for df.The inlet pressure period varied by about 20% and amplitude by about 6%, with no significant accuracy deterioration relative to baseline.Source: §5.2 · p. 16, Figure 8 · p. 17

6 Conclusions

The paper presents a SciML framework for automatically identifying the LC-DN-α stabilization parameter, using a nonlinear model of problem descriptors. The model generalizes across considered tests and moderate loading and geometry variations, but performance degrades with significant outlet-condition changes.

6 Conclusions

  • The proposed framework uses SciML to automatically identify the stabilization parameter α for the LC-DN-α scheme.The authors identify nonlinear relationships among physical, numerical, and algebraic features and α, and use ∆t, E, and ρs as network inputs.Source: §6 · pp. 18, 20
  • The model generalizes across considered test cases and provides robust estimates under moderate loading and geometry variations, but predictions degrade with significant outlet-boundary changes.The authors report that retraining is unnecessary for the tested inlet and stenosis variations, but recommend it for the absorbing-outlet case.Source: §6 · p. 20
  • Reliable parameter estimates under common geometry and inlet variations are presented as a step toward reducing computational demand in patient-specific computational hemodynamics.The authors frame clinical translation as a broader goal, not as an outcome demonstrated in this study.Source: §6 · p. 20
  • The work is a proof of concept for simplified 2D FSI with Stokes flows and linear-elastic structures; extensions to Navier–Stokes, nonlinear structures, and 3D remain future work.The authors caution that current inputs may not capture variability in more complex problems and that additional descriptors may be needed.Source: §6 · p. 20
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