Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

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

stat.MLAug 13, 2026

Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

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

Organizations: Aix Marseille Univ, CNRS, AMSE UMR 7316, Marseille, France · Aix Marseille Univ, CNRS, Centrale Marseille, IRPHE UMR 7342, Marseille, France

Abstract

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.

Figures & tables

Figure 1: Abdominal Aortic Aneurysm Idealized Geometry.

Figure 2: Flow rate waveforms $Q(t)$ derived from clinical data. (a) Waveforms grouped by patient source: FR1, FR2, and FR3 (five waveforms per group). (b) All 15 flow rate curves overlaid on a single plot, illustrating the full range of temporal and amplitude variability across patient groups. (c) Scatter plot of peak time versus peak flow rate amplitude for each waveform.

Architecture	Input	Trunk Basis Nature	Reference
Vanilla DeepONet	Single ( $f$ )	Static (fixed shapes)	Lu et al. [ 28 ]
Modified DeepONet	Single ( $f$ )	Adaptive (depends on $f$ )	Wang et al. [ 42 ]
Standard MIONet	Multi ( $f_{1},f_{2}$ )	Static (fixed shapes)	Jin et al. [ 21 ]
M 3 PI-DeepONet	Multi ( $f_{1},\ldots,f_{n}$ , $n$ arbitrary; illustrated below with $n=3$ )	Adaptive (depends on aggregated fusion of $f_{1},\ldots,f_{n}$ )	This work

Table 1: Architectural evolution of operator networks toward multi-input adaptive operators. The M 3 PI-DeepONet fills the previously unexplored combination of multi-input encoding with input-adaptive trunk basis functions.

Figure 3: Schematic of the M 3 PI-DeepONet in the physically natural three-branch instantiation ( $n=3$ ). Branches encode $v_{2}^{\{\text{inlet}\}(i)}$ , $p^{\{\text{outlet}\}(i)}$ , and $v_{2}^{\{\text{ic}\}(i)}$ at their sensor points; the trunk maps $(x_{1},x_{2},x_{3},t)$ . Modified architecture with Aggregated Injection (Figure 4 ) condition the coordinate basis on all three inputs jointly. The output layer produces $(\hat{\mathfrak{G}}_{\theta}^{(v_{1})},\hat{\mathfrak{G}}_{\theta}^{(v_{2})},\hat{\mathfrak{G}}_{\theta}^{(v_{3})},\hat{\mathfrak{G}}_{\theta}^{(p)})$ via output splitting. The training loop minimizes the total loss $\mathcal{L}(\theta)$ defined in Section 2.2.3 .

Figure 4: Modified architecture with Aggregated Injection, depicted for the three-branch instantiation ( $n=3$ ). The forward propagation in conjunction with the gating mechanism is modulated in such a way that each branch net uses its corresponding encoder $\mathbf{U}^{(\xi)}$ together with the trunk encoder $\mathbf{V}_{\text{trunk}}$ , while the trunk net uses its corresponding encoder $\mathbf{V}_{\text{trunk}}$ together the aggregated encoder $\mathbf{U}_{\text{agg}}$ , endowing the basis functions with their input-adaptive character. The * operation is the one defined in Eqs.( 14 )-( 15 ).

Figure 5: Overview of the dataset generation pipeline. Patient-specific flow rate waveforms

Q(t)

are converted into time-dependent inlet velocity profiles

\boldsymbol{v}^{\{\text{inlet}\}}(\boldsymbol{x},t)

. For each profile, a CFD simulation produces snapshots at discrete time steps

t_{0},t_{1},\ldots,t_{100}

. At every time step, the resulting data is organized into four geometrically distinct strata: (1) Inlet, (2) Outlet, (3) Wall, and (4) Volume.

Figure 6: Spatio-temporal data splitting strategy for a single velocity profile-dependent dataset. The full set of 101 recorded time steps (

t_{0}

–

t_{100}

) is shown on the left. On the right, the dataset is partitioned along both axes: temporally into 51 training steps, a 5-step validation buffer, and 45 test steps; spatially into 68% training, 2% validation, and 30% test. The model trains on the lower-left block. The remaining blocks define the three evaluation scenarios: Spatial Test , Temporal Test , and Hard Test .

Figure 7: Data : Mimicking an external sparse data source via random sampling within the Volume region. The temporal indices $\tilde{t}_{i}$ correspond directly to the 51 training time steps, and the $0.3\%$ spatial sampling is drawn once and reused identically across all these steps.

		Full Test
No. Branches	Branch Inputs	MoV (%)	Pressure (%)	Training Runtime (h)	#Params
1	$v_{2}^{\{\text{inlet}\}}$	27.61 ± 0.27	12.16 ± 3.44	2.33	2.4990e+5
1	$p^{\{\text{outlet}\}}$	26.39 ± 0.58	9.82 ± 2.76	2.38	2.4990e+5
1	$v_{2}^{\{\text{ic}\}}$	14.20 ± 0.88	8.51 ± 0.84	2.3	2.2720e+5
2	$v_{2}^{\{\text{inlet}\}}$ , $p^{\{\text{outlet}\}}$	23.44 ± 0.89	11.94 ± 2.28	2.56	4.2860e+5
2	$v_{2}^{\{\text{inlet}\}}$ , $v_{2}^{\{\text{ic}\}}$	11.10 ± 0.23	9.17 ± 2.34	2.57	4.0590e+5

Table 2: Branch input study.

L^{2}

-relative errors (%) for Magnitude of Velocity (MoV) and Pressure follow the across-waveforms conventions, averaging across the three held-out waveforms used for testing. Runtime denotes the total training wall-clock time. #Params represents the total number of trainable parameters

|\theta|

of the network.

Figure 8: Impact of branch input selection on prediction accuracy. Top panel: magnitude of velocity; bottom panel: pressure. For each branch configuration, grouped bars show the $L^{2}$ -relative error (%) follow the across-waveform convention for the 12 training waveforms. In addition, the three test protocols (Spatial, Temporal, and Hard) are shown, with error bars indicating the standard deviation across waveforms. Configurations are separated by dashed vertical lines into single-branch (left), two-branch (center), and three-branch (right) groups.

Config	AI	ED	DW	KI	MoV (%)	Pressure (%)	Training Runtime (h)	#Params
					Full Test
A1	✗	✗	✗	✗	11.10 ± 0.23	9.17 ± 2.34	2.58	4.0590e+5
A2	✓	✗	✗	✗	7.08 ± 1.19	10.65 ± 3.34	8.7	5.6940e+5
A3	✓	✓	✗	✗	6.62 ± 1.11	5.48 ± 1.05	8.6	5.6940e+5
A4	✓	✓	✓	✗	5.50 ± 0.75	5.35 ± 0.57	8.65	5.6940e+5
A5	✗	✓	✓	✓	10.92 ± 0.65	6.15 ± 1.48	3.26	4.0590e+5

Table 3: Ablation study isolating the contribution of each architectural and training component. All models use 2 branches (

v_{2}^{\{\text{inlet}\}}

v_{2}^{\{\text{ic}\}}

). Errors follow the across-waveforms conventions, averaging across the three held-out waveforms. AI = Aggregated Injection (modified arch); ED = Exponential Decay; DW = data-weighted fixed loss setting (

\lambda_{\text{data}}=100

, all other weights equal to one); KI = Kaiming Initialization.

Branches	Branch Inputs	MoV (%)	Pressure (%)	Training Runtime (h)	#Params
		Full Test
2	$v_{2}^{\{\text{inlet}\}}$ , $v_{2}^{\{\text{ic}\}}$	3.37 ± 0.06	4.25 ± 1.26	8.65	5.6940e+5
3	$v_{2}^{\{\text{inlet}\}}$ , $p^{\{\text{outlet}\}}$ , $v_{2}^{\{\text{ic}\}}$	4.21 ± 1.28	4.69 ± 1.27	11.68	8.4600e+5
4	$v_{2}^{\{\text{inlet}\}}$ , $p^{\{\text{outlet}\}}$ , $v_{2}^{\{\text{ic}\}}$ , $p^{\{\text{ic}\}}$	3.63 ± 0.67	3.69 ± 1.08	14.48	1.0772e+6

Table 4: M 3 PI-DeepONet performance for the optimized training configuration. Errors follow the across-waveforms conventions, averaging across the three held-out waveforms.

			Magnitude of Velocity			Pressure
Patient Group	Waveform	Split	Spatial Test	Temporal Test	Hard Test	Spatial Test	Temporal Test	Hard Test
FR1	V1	Training set	4.07	4.01	4.04	0.33	4.09	4.43
FR1	V3	Training set	2.74	2.69	2.72	0.25	3.88	4.26
FR1	V4	Training set	3.22	3.16	3.20	0.27	3.99	4.36
FR1	V5	Training set	2.76	2.71	2.74	0.24	3.89	4.26
FR1	V2	Test set	3.24	3.19	3.22	0.73	4.04	4.38

Table 5: Per-waveform

L^{2}

-relative error (%) for the M 3 PI-DeepONet with the two-branch configuration

(v_{2}^{\{\text{inlet}\}},\,v_{2}^{\{\text{ic}\}})

. Results are grouped by patient source and train/test assignment. The bottom three rows report the across-waveform “

\mu\pm\sigma

” over the training set, the test set, and all 15 waveforms.

Figure 9: Qualitative comparison between the CFD ground truth (left), M 3 PI-DeepONet prediction using the two-branch configuration $(v_{2}^{\{\text{inlet}\}},\,v_{2}^{\{\text{ic}\}})$ (center), and pointwise absolute error (right) for the held-out test waveform FR1V2 at peak systole ( $t=0.16$ s). Top row: magnitude of velocity; bottom row: pressure.

Figure 10: Temporal evolution of the predicted (dashed) and ground truth (solid) fields at probe point P 1 , located at the center of the aneurysm bulge (coordinates $(0,\,0.0617,\,0)$ ), for the three held-out test waveforms, one per patient group (FR1 $\cdot$ V2, FR2 $\cdot$ V3, FR3 $\cdot$ V2). (a) Magnitude of velocity. (b) Pressure. The two-branch M 3 PI-DeepONet accurately tracks both the systolic peak and diastolic recovery phases. Notice that the inset identifies both probe locations, P 1 and P 2 , whereas the temporal curves shown here correspond only to P 1 .

			Velocity ( $v$ ) [m/s]		Pressure ( $p$ ) [Pa]
Probe	Patient	Test Waveform	MAE	Peak	MAE	Peak
P 1 (bulge centre)	FR1	V2	0.0045	0.0117	0.852	6.994
P 1 (bulge centre)	FR2	V3	0.0019	0.0057	1.253	9.136
P 1 (bulge centre)	FR3	V2	0.0038	0.0097	0.730	4.304
Mean $\pm$ Std (P 1 )			0.0034 $\pm$ 0.0014	0.0091 $\pm$ 0.0031	0.945 $\pm$ 0.274	6.811 $\pm$ 2.421
P 2 (off-axis)	FR1	V2	0.0036	0.0090	0.779	6.040

Table 6: Quantitative evaluation of the absolute errors at the two probe points shown in the inset of Figure 10 : P 1 at the bulge centre,

(0,\,0.0617,\,0)

, and P 2 at the off-axis location

(0.0,\,0.0667,\,0.015)

. Results are reported for the three held-out test waveforms using the two-branch configuration.

Method	Stage	Time	Hardware
CFD (ANSYS Fluent)	Resolved retained cycle	$\sim$ 12 h	13th Gen Intel(R) Core(TM)
			i9-13900K (3.00 GHz) RAM 128 GB
M 3 PI-DeepONet	Training (one-time)	8 h - 15 h	NVIDIA RTX A6000
M 3 PI-DeepONet	Full DS Inference * (per waveform)	$\sim$ 20 min	NVIDIA RTX A6000
Retained-cycle inference speedup		$\boldsymbol{\sim}$ 36 $\times$

Table 7: Computational cost comparison between the CFD solver and the M 3 PI-DeepONet. * Full DS inference denotes evaluation over the retained dataset representation of the converged cardiac cycle, discretized into

101

uniformly spaced snapshots.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11: Cross-waveform dataset splitting process. The 15 velocity profiles are organized into three patient groups (FR1, FR2, FR3), each containing five variants (V1-V5). One specific variant from each group (V2 for FR1 and FR3, and V3 for FR2) is held out as an entirely unseen test set (3 profiles), evaluating the operator’s generalization to new waveforms. The remaining 12 profiles form the training set, within which the spatio-temporal split of Figure 6 applies.

Patient Group	Waveform	$V$ [m/s]	$Re$	$\alpha$
FR1	V1	0.388	1038	13.0
FR1	V2	0.331	884	13.0
FR1	V3	0.365	976	13.0
FR1	V4	0.340	910	13.0
FR1	V5	0.281	751	13.0
FR2	V1	0.346	926	13.0

Appendix

Table 8: Physical parameters and dimensionless numbers for the 15 waveforms.

Figure 12: Cardiac-cycle filmstrip of the magnitude of velocity for the held-out test waveform FR1V2. Top inset: inlet flow-rate waveform with four characteristic phases (early systole, acceleration, peak systole, deceleration) and the $14$ sampled time instants indicated by vertical markers. Each column corresponds to one of these instants. Top row: CFD ground truth; middle row: M 3 PI-DeepONet prediction; bottom row: pointwise absolute error $\lvert v(\boldsymbol{x})-\hat{v}_{\theta}(\boldsymbol{x})\rvert$ . The maps are evaluated on the $x_{3}=0$ probe plane.

Figure 13: Corresponding cardiac-cycle filmstrip of the pressure field. Similar to Figure 12 , this strip shows the evolution of the pressure field for the held-out test waveform FR1V2, on the same $14$ time instants and the $x_{3}=0$ plane. Top row: CFD ground truth; middle row: M 3 PI-DeepONet prediction; bottom row: pointwise absolute error $\lvert p(\boldsymbol{x})-\hat{p}_{\theta}(\boldsymbol{x})\rvert$ .

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