Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

Oscar L. Cruz-Gonzalez; Valérie Deplano; Badih Ghattas

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Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

Oscar L. Cruz-Gonzalez, Valérie Deplano, Badih Ghattas

arXiv:2608.13629v1Aug 13, 2026stat.MLcs.LGphysics.flu-dyn

TL;DR

Patient-specific, time-resolved hemodynamic data are difficult to obtain clinically, motivating predictive tools for AAA assessment. This paper introduces M3PI-DeepONet with Aggregated Injection for unsteady 3D AAA flow prediction, achieving 3.37% MoV, 4.25% pressure error, and approximately 36× faster inference than reference CFD.

Problem
High-fidelity, time-resolved AAA hemodynamic data remain difficult to obtain clinically, while diameter-based criteria insufficiently discriminate unfavorable evolution.
Method
M3PI-DeepONet learns unsteady 3D Navier–Stokes flow operators using multi-branch inputs and learned Aggregated Injection to create an input-adaptive trunk basis.
Results
3.37% MoV and 4.25% pressure error were achieved in the full configuration, with approximately 36× retained-cycle inference speedup versus reference CFD.
Takeaways & Limitations
The framework moves toward rapid hemodynamic assessment through conditional reconstruction that is substantially faster than repeated CFD simulations.
Takeaways & Limitations
The study uses one idealized rigid AAA geometry and conditioning inputs partly known from retained-cycle CFD, so patient-specific generalization remains un demonstrated.

Abstract

from arXiv · show

Clinically actionable, patient-specific hemodynamic assessment, specifically wall shear stress, vortex structure and pressure distributions, is critical for determining risky or unfavorable evolution in Abdominal Aortic Aneurysms (AAA). While Physics-Informed Deep Operator Networks (PI-DeepONets) show promising results in complementing established 5 tools such as Computational Fluid Dynamics (CFD), a persistent architectural challenge remains for complex 3D flows. In this direction, we propose a Modified Multi-Input Multi-Output PI-DeepONets (M3PI-DeepONet) designed for predicting unsteady flows in an idealized AAA geometry. Central to our model is the Aggregated Injection strategy, where latent representations from multiple input branches are fused prior to trunk injection, allowing the coordinate basis to adapt to multiple physical constraints. To the best of our knowledge, this is the first architecture to combine the layer-wise gating mechanism with a multi-branch operator-network topology, yielding an input-adaptive trunk basis. Additionally, we integrate the 3D Navier-Stokes equations as governing physical laws, so the model is trained based on physics-informed residuals, initial and boundary conditions, and only 0.3% of the labeled internal data together with the selected branch-conditioning signals. The M3PI-DeepONet simultaneously predicts unsteady 3D flow velocity and pressure fields with an average relative L2 velocity error below 4% and pressure error around 5% while achieving a conservative retained-cycle inference speedup of approximately 36x compared to reference CFD simulations once the branch inputs used for conditioning are available. This work advances the application of deep learning in cardiovascular disease modeling, marking step toward real-time, non-invasive clinical diagnostics.

1. Introduction

The introduction motivates fast, patient-specific AAA hemodynamic assessment beyond diameter-based criteria and positions M3PI-DeepONet as a multi-input, physics-informed operator-learning approach for unsteady 3D flow. It combines adaptive trunk representations with multiple branch inputs while targeting computational efficiency relative to CFD.

Clinical motivation: Hemodynamic markers including wall shear stress, pressure, and vortex dynamics influence AAA evolution, whereas diameter-based clinical criteria are insufficient discriminants.AAA rupture is associated with high mortality, motivating richer hemodynamic assessment.
Clinical motivation: 4D flow MRI has limited spatial and temporal resolution near vessel walls, while CFD-based assessment can impose prohibitive computational costs.These limitations complicate accurate wall-location and wall-shear-stress estimation and motivate efficient surrogate modeling.
Operator learning: DeepONets learn generalized mappings between infinite-dimensional function spaces, enabling rapid predictions across varying source terms, geometries, or boundary conditions.PI-DeepONets embed physical laws into the loss function, reducing total dependence on labeled data.
Architectural motivation: A static trunk basis can restrict coupled velocity–pressure modeling because it represents outputs with the same coordinate-dependent functions independently of branch conditions.Branch-adaptive basis functions are presented as a natural way to improve representation of coupled flow fields.
Contributions: M3PI-DeepONet extends layer-wise gating from a single-branch architecture to the multi-branch MIONet topology for unsteady flow prediction in an idealized 3D AAA geometry.The work also evaluates unsteady 3D velocity and pressure prediction accuracy and computational efficiency relative to CFD.
Scope and limitations: The benchmark establishes time-dependent operator learning for pathological vascular flow while acknowledging that the idealized geometry omits full in vivo AAA complexity, including fluid–structure interaction effects.The authors frame this benchmark as a prerequisite for fast, patient-specific hemodynamic modeling.

2. Methodology · 2.1. Problem Formulation and CFD Ground Truth

The methodology formulates unsteady blood flow in an idealized three-dimensional AAA using incompressible Newtonian Navier–Stokes equations and boundary conditions, then generates periodic CFD ground truth across clinically derived flow waveforms for operator learning. The framework represents velocity and pressure as spatiotemporal fields and uses the converged cardiac cycle as the retained reference state.

2.1.1. Unsteady AAA flow problem formulation: The computational domain B ⊂ R^3 represents the lumen of an idealized three-dimensional AAA, bounded by inlet, outlet, and rigid arterial-wall surfaces.The independent variables are x_1, x_2, x_3, and t; dependent variables are the three velocity components and pressure.
2.1.1. Unsteady AAA flow problem formulation: Incompressible Newtonian 3D Navier–Stokes equations govern the blood-flow momentum and mass conservation.The Newtonian assumption is considered valid for large arteries with sufficiently high shear rates.
2.1.1. Unsteady AAA flow problem formulation: The problem is closed with a patient-derived, time-dependent inlet velocity profile, no-slip rigid-wall conditions, and a standard fully developed outflow condition.The inlet profile is derived from real patient flow rate Q(t) using the Womersley solution.
2.1.1. Unsteady AAA flow problem formulation: The target hemodynamic regime is periodic, satisfying v(x, t) = v(x, t + T ) and p(x, t) = p(x, t + T ) across cardiac cycles.This periodic phase state is reached because the inlet velocity profile is periodic with physical period T.
2.1.2. Ground-truth CFD generation: 15 distinct flow-rate waveforms Q(t) expand the CFD training distribution, spanning three patient groups—FR1, FR2, and FR3—with five waveforms per group.The groups share temporal profiles within each group but differ in peak amplitude.
2.1.2. Ground-truth CFD generation: Four cardiac cycles of T = 1 s were simulated with 1000 computational time steps per cycle (∆t = 0.001 s), retaining the converged fourth cycle as the reference state.The fourth cycle satisfies the periodicity condition and supplies the t = 0 velocity and pressure snapshot used by the operator-learning framework.
2.2.1. Operator Learning Foundation: Operator learning avoids retraining for changed problem parameters by learning mappings between infinite-dimensional function spaces rather than a single PDE solution.This enables instant predictions for entire families of PDEs.

2.2. The M 3PI-DeepONet Formulation

The M3PI-DeepONet learns a nonlinear operator mapping selected input functions to full spatio-temporal Navier–Stokes fields in an idealized AAA geometry. Its multi-branch Aggregated Injection architecture conditions trunk coordinate bases on joint input context while using multiple outputs for the four physical fields.

2.2. The M 3PI-DeepONet Formulation: The general operator-learning framework maps input functions to PDE solution functions while enforcing interior and boundary residual operators.Standard DeepONet approximates this map with branch and trunk subnetworks, whereas PI-DeepONet adds PDE residuals, boundary conditions, and initial conditions to training.
2.2.2. Proposed Architecture: The formulation learns an operator G mapping selected input functions to the full spatio-temporal solution of incompressible Navier–Stokes equations in an idealized AAA geometry.The active inputs are selectable information channels rather than a fixed set of problem parameters.
2.2.2. Proposed Architecture: The selected channels are information supplied to the operator, not necessarily prescribed boundary or initial conditions; v2 and p are physically motivated choices that could be expanded.Additional velocity components or informative quantities could in principle be included in S.
2.2.2. Proposed Architecture: Inputs extracted from the retained fourth CFD cycle represent measurement-conditioned or data-assimilation settings rather than surrogates driven solely by prescribed inlet waveforms.In prospective clinical or CFD-free workflows, these channels would need to come from external measurements.
2.2.2. Proposed Architecture: The Aggregated Injection strategy combines layer-wise gating with multi-input topology, giving the trunk basis an input-adaptive character.The design extends a single-branch gating mechanism to multiple branches and avoids biasing the coordinate basis toward one input channel.
2.2.2. Proposed Architecture: The architecture uses n independent input branches, a spatio-temporal trunk, and Aggregated Injection to condition coordinate bases on all active inputs jointly.The formulation is written for arbitrary n; the three-branch configuration is illustrative rather than architectural necessity.
2.2.2. Proposed Architecture: Each branch encodes one active input function at fixed sensor points into a latent embedding, while the trunk maps (x1, x2, x3, t) into a basis embedding.The trunk conditioning uses the joint context of all active input branches rather than any single channel.
2.2.2. Proposed Architecture: The model predicts four scalar fields—v1, v2, v3, and p—by partitioning q trunk-branch interaction modes into 4 disjoint output slices.This multiple-output extension replaces four independent networks with one network whose interaction modes are split among output components.

2.3. Dataset Construction

The dataset is built from precomputed CFD snapshots organized by geometry and split jointly across space and time. Training uses sparse supervised targets alongside physics collocation points, while branch inputs remain architecturally variable.

Data organization: CFD snapshots are extracted at each recorded time step and organized into Inlet, Outlet, Wall, and Volume strata containing coordinates and flow variables.Each stratum provides access to (x1, x2, x3, v1, v2, v3, p), and the fixed dataset is not dynamically resampled during training.
Spatio-temporal splitting: The spatio-temporal split uses 51 training steps, a 5-step validation buffer, and 45 test steps, with spatial proportions of 68%, 2%, and 30%.The resulting Spatial, Temporal, and Hard Tests separately assess spatial, temporal, and joint spatio-temporal generalization.
Loss sampling: Physics collocation points are sampled across the full cardiac-cycle time range, whereas supervised loss points are restricted to the Train block.This distinction applies because the split constrains Ldata, Linlet, Loutlet, Lwall, and Lic, but not Lphy.
Sparse supervision: 0.3% of available Volume points are randomly retained as sparse interior sensors, with the same spatial sample reused across all 51 training steps.These points provide known velocity and pressure targets through Ldata.
Target and branch-input roles: Loss-target data remains fixed while stored fields can alternatively serve as branch-input functions, so changing active branches does not change supervision.For example, inlet velocity can serve both roles, outlet pressure only as a possible branch input, and wall velocity only as a loss target.

3. Results

Results show that initial-condition conditioning and multi-input modeling are important for accurate coupled velocity–pressure prediction, while the full M3PI-DeepONet recipe substantially improves performance. The optimized two-branch model offers the strongest accuracy–complexity balance and generalizes well to unseen waveforms, although pressure remains more variable across test conditions.

Input-branch analysis: 14.20% MoV and 8.51% pressure errors make the initial-condition branch more informative than inlet velocity or outlet pressure alone.All two-branch combinations including the initial condition outperform single-branch models, while the pair omitting it reaches 23.44% MoV error.
Input-branch analysis: 11.10% velocity and 9.17% pressure errors identify the inlet-plus-initial-condition pair as the best two-branch configuration under the standard architecture.The three-branch model instead has a higher 16.96% MoV error, indicating difficulty fusing heterogeneous inputs without the modified architecture and Aggregated Injection.
Ablation study: 7.08% MoV error and 36.21% relative improvement result from adding the modified architecture with Aggregated Injection to baseline A1, while pressure is 10.65%.A1 has 11.10% MoV and 9.17% pressure errors; removing the modified architecture from the otherwise enhanced recipe raises MoV error to 10.92%.
Ablation study: 5.50% MoV and 5.35% pressure errors are achieved after fixed loss weights, following exponential decay results of 6.62% and 5.48%.The final A6 recipe combines physics-informed loss, Aggregated Injection, exponential decay, fixed weights, and Kaiming Uniform initialization.
Optimized model comparison: 3.37±0.06% MoV error, 4.25 ± 1.26% pressure error, and 8.65 h training runtime make the optimized two-branch model the best accuracy–complexity trade-off.The four-branch extension achieves the lowest pressure error, 3.69±1.08%, but requires the largest model and longest runtime.
Generalization and field evaluation: 3.63% test-set mean MoV Hard error and 5.15% test pressure Hard error remain close to training values of 3.12% and 5.11%, demonstrating generalization to unseen waveforms.Pressure errors are approximately 3.6–4.4% for FR1 and FR3 but approximately 7.3–7.4% for FR2, whose sharper waveform variations create steeper pressure gradients.

4. Discussion and conclusions

The M3PI-DeepONet combines multi-branch operator learning with Aggregated Injection to predict unsteady 3D hemodynamics in an idealized AAA geometry. Results indicate improved accuracy–complexity trade-offs and approximately 36× retained-cycle inference acceleration, while limitations remain in conditioning requirements, temporal resolution, and generalization beyond the benchmark geometry.

Contributions: M3PI-DeepONet predicts unsteady 3D hemodynamics in an idealized abdominal aortic aneurysm geometry using the Aggregated Injection strategy.The strategy fuses latent representations from active input branches before trunk injection, combining layer-wise gating with a multi-branch operator topology.
Key findings: The initial-condition branch v{ic}2 was the most informative candidate input, while the standard architecture degraded when integrating the three-branch natural triple.The pair (v{inlet}2, v{ic}2) gave the best trade-off in the standard MI-MO-PI-DeepONet.
Key findings: The two-branch configuration provided the best accuracy–complexity trade-off, whereas adding p{ic}2 achieved the lowest pressure error at increased computational cost.The two-branch model had the lowest MoV error with the fewest parameters and shortest training time.
Computational efficiency: 36× retained-cycle inference speedup was achieved relative to the reference CFD solver once the trained operator and conditioning inputs were available.The framework therefore reduces prediction time but does not eliminate the offline CFD ground-truth bottleneck or all CFD-derived conditioning requirements.
Limitations: ∼7% Hard test pressure error in the FR2 patient group indicates that sharper systolic peaks and greater temporal variability challenge the model’s temporal resolution.The interpretation identifies temporal resolution as the primary bottleneck for the most complex physiological inputs.
Limitations and future work: Generalization remains un demonstrated beyond a single idealized AAA geometry with a rigid vessel wall and measurement-conditioned branch information.Future work proposes geometry encoding, compliant-wall mechanics, and alternative initialization or measurement strategies to broaden applicability.

CRediT authorship contribution statement

OLCG led conceptualization, methodology, software, visualization, formal analysis, and the original draft. BG and VD contributed to funding, conceptualization, methodology, formal analysis, and manuscript review, with VD also constituting the CFD database.

OLCG contributed conceptualization, methodology, software, visualization, formal analysis, and writing the original draft.
BG contributed funding acquisition, conceptualization, methodology, formal analysis, and writing review and editing.
VD contributed funding acquisition, conceptualization, methodology, formal analysis, CFD database constitution, and writing review and editing.

Appendix A. Training Protocol

Training used fixed architecture and optimization settings, structured temporal and cross-waveform data partitions, and profile- and point-level mini-batching. Inputs were nondimensionalized before training, with dimensional fields restored at inference.

Dimensionless approach: Coordinates, velocities, and pressures were scaled using the characteristic quantities from the dimensionless analysis before training, then inverse-transformed at inference.The CFD ground truth was generated and stored in dimensional variables.
Temporal partitioning: Training used even-indexed snapshots t0, t2, . . . , t100, while odd-indexed snapshots t1, t3, . . . , t99 formed validation and test blocks.t100 was retained only as the periodic closure of t0.
Network sizing and optimizer: All models used 4 hidden layers of 100 neurons with tanh activations and were trained for 200,000 Adam iterations at η = 10^-3.The baseline used Xavier Uniform initialization, with β1 = 0.9 and β2 = 0.999.
Cross-waveform partitioning: 12 profiles were used for training and 3 held-out profiles for testing, with V2 held out from FR1 and FR3 and V3 from FR2.The split covers 15 velocity profiles across three patient groups and evaluates generalization to unseen waveforms.
Profile-level batching: Each iteration sampled 6 of the 12 training profiles, promoting waveform diversity while reducing memory requirements.Profile-level stochastic sampling exposed the operator to varied waveform conditions throughout training.
Loss-term weighting: The composite loss used either unit weights for all terms or λdata = 100 with all remaining weights equal to one.No adaptive loss balancing was used.

Appendix B. Dimensionless form of NSE for the unsteady AAA idealized model

Appendix B derives the dimensionless unsteady Navier–Stokes equations for the idealized AAA model using inlet velocity, radius, and pulsatile-flow frequency scales. The resulting formulation identifies Womersley-number effects and summarizes dimensionless parameters across 15 flow waveforms.

Reference scales: The nondimensionalization uses maximum inlet velocity V, inlet radius R = R_inlet, and angular frequency ω = 2π/T as representative physical scales.T denotes the cardiac period.
Dimensionless formulation: The derivation rescales spatial gradients, Laplacians, and time derivatives before rewriting the Navier–Stokes equations with kinematic viscosity ν_f = μ_f/ρ_f.The transformations include ∇* = R∇, (∇*)^2 = R^2∇^2, and ∂/∂t = ω∂.
Womersley regime: When α ≫1, oscillatory inertia dominates viscous effects within each cycle, describing the regime relevant to large-artery pulsatile flows.The relative magnitude of the unsteady term to the convective term is governed by the Womersley number α.
Study parameters: 15 flow rate waveforms share R = 0.010065 m, ρ_f = 1060 kg/m^3, μ_f = 0.00399 kg/(m s), and T = 1 s, yielding identical α ≈13.0.The Reynolds number varies with each waveform’s peak inlet velocity.

Appendix C. Cardiac-cycle filmstrip

Appendix C compares CFD ground truth, M3PI-DeepONet predictions, and pointwise absolute errors across 14 cardiac-cycle instants for held-out waveform FR1V2. Velocity and pressure evolve through the cycle with corresponding filmstrips, while other held-out waveforms show qualitatively similar behavior.

Filmstrip overview: 14 cardiac-cycle instants are evaluated for held-out waveform FR1V2 using CFD ground truth, M3PI-DeepONet predictions, and pointwise absolute error.The inlet waveform marks early systole, acceleration, peak systole, and deceleration.
Velocity evolution: Velocity magnitude filmstrips show the CFD ground truth, M3PI-DeepONet prediction, and pointwise absolute error at each sampled instant.The maps are evaluated on the x3 = 0 plane.
Pressure evolution: Pressure filmstrips similarly show ground truth, prediction, and pointwise absolute error across the same 14 time instants and x3 = 0 plane.Equivalent filmstrips for held-out waveforms FR2V3 and FR3V2 display qualitatively similar behavior and are omitted for conciseness.