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Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

This paper introduces a novel Modified Multi-Input Multi-Output Physics-Informed DeepONet (M3PI-DeepONet) featuring an Aggregated Injection strategy and layer-wise gating to accurately and efficiently predict unsteady 3D hemodynamics in abdominal aortic aneurysms with minimal labeled data, achieving significant speedups over traditional CFD simulations for potential real-time clinical diagnostics.

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

Published 2026-08-17

📖 6 min read🧠 Deep dive

CC0 1.0

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

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a river will flow through a winding canyon. In the real world, rivers are messy; they swirl, crash against rocks, and change speed depending on the rain. In the world of medicine, our blood vessels are like those rivers, and sometimes they develop weak spots called aneurysms—bulges that can burst if the pressure gets too high. Doctors need to know exactly how the blood is pushing and swirling inside these bulges to decide if a patient needs surgery.

To get this information, scientists usually use a super-powerful computer program called Computational Fluid Dynamics (CFD). Think of CFD as a high-tech, slow-motion movie camera that simulates every drop of blood. The problem is, making this movie takes a long time—sometimes days on a supercomputer. It's like trying to predict the weather by simulating every single molecule of air; it's accurate, but it's too slow to help a doctor make a quick decision in a hospital.

Recently, scientists have started using "AI" (Artificial Intelligence) to speed things up. Instead of simulating every drop, the AI learns the rules of physics and tries to guess the answer instantly. One popular type of AI is called a "DeepONet." You can think of a standard DeepONet as a very smart student who memorizes a specific textbook. If you ask it a question from that textbook, it answers perfectly. But if you give it a slightly different question (like a new patient with a different heart rate), it gets confused and has to start studying all over again. This paper introduces a new, upgraded version of this AI student that doesn't just memorize; it learns to adapt its brain to new situations instantly.

The Paper: Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

This paper is about teaching a super-smart AI to predict how blood flows through a fake, idealized version of a dangerous abdominal aortic aneurysm (a bulge in the main artery of the belly). The goal is to create a tool that can tell doctors the speed of the blood and the pressure on the artery walls in seconds, rather than the 12 hours it currently takes for a traditional computer simulation.

The Problem with the Old AI
Imagine you are trying to describe a dance to a friend. If you only tell them the music (the inlet flow), they might guess the moves. If you only tell them the ending pose (the outlet pressure), they might guess differently. But if you tell them both, plus how the dance started (the initial state), they can figure out the whole routine perfectly.

Previous AI models were like a friend who could only listen to one thing at a time. If you gave them two or three clues (like the inlet flow and the initial state), they got confused because their "brain" (the part that generates the answer) was rigid. It was like trying to wear a pair of shoes that only fit one specific foot size, no matter how you tried to stretch them. The paper argues that simply adding more inputs to the old AI didn't work; it actually made the predictions worse because the AI couldn't mix the different clues together properly.

The New Solution: The "Aggregated Injection" Strategy
The authors created a new AI architecture called M3PI-DeepONet. The secret sauce here is something they call the "Aggregated Injection" strategy.

Think of the AI as a chef making a soup.

The Branches: These are the ingredients. In this case, the ingredients are different pieces of information about the blood flow (like the speed at the entrance, the pressure at the exit, and how the blood was moving at the very start of the heartbeat).
The Trunk: This is the pot where the soup is cooked. In old AI models, the pot was rigid. It had a fixed shape, so no matter what ingredients you threw in, the soup always tasted the same.
The Innovation: The authors added a special "blender" (the Aggregated Injection) that mixes all the ingredients together before they go into the pot. This changes the shape of the pot itself! Now, the pot adapts to the specific mix of ingredients. If you add a spicy ingredient, the pot stretches to handle the heat. If you add a watery one, the pot shrinks.

This allows the AI to create an "Input-Adaptive Basis." In plain English, the AI changes its internal map of the world based on the specific clues you give it. It stops being a rigid robot and starts being a flexible problem-solver.

What They Found
The team tested this new AI on a computer model of an aneurysm. They fed it data from 15 different heartbeats (waveforms) and asked it to predict the blood flow for new, unseen heartbeats.

It works better with more clues: They found that giving the AI just one clue (like the inlet flow) resulted in a messy prediction with about a 27% error. But when they gave it two specific clues—the inlet flow and the initial state of the blood—the error dropped dramatically to about 3.4% for speed and 4.2% for pressure.
The "Blender" is key: When they tried to use three clues without their new "blender" strategy, the AI got confused and the error went back up. But with the Aggregated Injection, even using three or four clues worked great. This proves that the way they mixed the information was the real breakthrough.
Speed is the winner: The most exciting part is the speed. A traditional computer simulation took about 12 hours to calculate the flow for one heartbeat cycle. The new AI, once it was trained, could do the same job in about 20 minutes. That is a 36 times speedup.

What It Can't Do Yet
The paper is very honest about its limits. The AI was trained on a "perfect" or "idealized" shape of an aneurysm. Real human arteries are bumpy, curved, and unique to every person. This AI doesn't know how to handle those real-world wrinkles yet. Also, the AI needs some specific starting information (like the blood speed at the very beginning of the heartbeat) to make its prediction. In a real hospital, getting that exact starting data might be tricky without doing a scan first.

The Bottom Line
This paper doesn't claim to have solved the problem of predicting aneurysms for every patient tomorrow. Instead, it shows a powerful new way to build AI that learns the laws of physics and adapts to new situations instantly. By mixing different clues together before feeding them into the AI's brain, they created a model that is much faster and more accurate than previous attempts. It's a big step toward a future where doctors might be able to run a "what-if" simulation on a patient's heart in the time it takes to brew a cup of coffee, helping them decide on life-saving treatments much faster.

Technical Summary: Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

Problem Statement
The clinical assessment of Abdominal Aortic Aneurysms (AAA) relies heavily on hemodynamic markers such as wall shear stress (WSS), vortex structures, and pressure distributions to predict rupture risk. While Computational Fluid Dynamics (CFD) provides high-fidelity solutions, it is computationally prohibitive for routine clinical use, often requiring hours to days per patient. Conversely, non-invasive imaging (e.g., 4D flow MRI) lacks the spatial and temporal resolution necessary for accurate wall shear stress estimation. Physics-Informed Neural Networks (PINNs) and Deep Operator Networks (DeepONets) offer a potential bridge, yet existing architectures face challenges when applied to complex, unsteady 3D flows. Specifically, standard multi-input operator networks often utilize static coordinate bases that fail to adapt to varying boundary conditions, and single-input architectures struggle to fuse heterogeneous physical constraints effectively.

Methodology
The authors propose a Modified Multi-Input Multi-Output Physics-Informed Deep Operator Network (M3PI-DeepONet) designed to predict unsteady 3D velocity and pressure fields in an idealized AAA geometry. The framework is governed by the 3D incompressible Navier-Stokes equations.

Architecture: The model integrates two distinct operator learning paradigms:
1. Layer-wise Gating: Adapted from Wang et al. [32], this mechanism allows the network to dynamically morph its internal representations based on input conditions.
2. Multi-Branch Topology: Adapted from Jin et al. [33] (MIONet), this allows for multiple input functions (e.g., inlet velocity, outlet pressure, initial conditions).
Aggregated Injection Strategy: The core innovation is a novel strategy where latent representations from multiple active input branches are fused via a learnable weighted sum before being injected into the trunk network. Unlike standard MIONet, where the trunk basis remains static regardless of branch inputs, this approach creates an input-adaptive coordinate basis. The trunk network's basis functions $\tau_k(x, t)$ become dependent on the joint context of all active branches, allowing the model to better capture the coupled physics of velocity and pressure.
Physics-Informed Training: The network is trained using a composite loss function that minimizes:
- Data loss (sparse internal measurements, only 0.3% of volume data).
- Boundary and initial condition losses (inlet velocity, outlet pressure, no-slip wall, initial state).
- Physics loss (residuals of the dimensionless unsteady Navier-Stokes equations).
  The model requires no labeled internal data for the physics loss, relying instead on the governing equations to regularize the solution.

Key Contributions

Novel Architecture: This work presents the first peer-reviewed application combining a layer-wise gating mechanism with a multi-branch MIONet topology. It introduces the Aggregated Injection strategy to resolve the dimensional challenge of conditioning a trunk network on multiple heterogeneous inputs.
Unsteady 3D Hemodynamics: The framework extends operator learning from steady or reduced-order settings to unsteady, three-dimensional flows within a pathological vascular geometry.
Input-Adaptive Basis: By fusing branch representations prior to trunk injection, the model shifts from a static coordinate basis to an input-adaptive one, significantly improving the representation of coupled flow fields.

Results
The model was evaluated on an idealized AAA geometry using 15 distinct flow rate waveforms derived from clinical data (12 for training, 3 for testing).

Accuracy: The optimized M3PI-DeepONet (using a two-branch configuration of inlet velocity and initial state velocity) achieved an average relative $L_2$ $L_{2}$ error of 3.37% for velocity magnitude and 4.25% for pressure on unseen waveforms.
- Ablation studies demonstrated that the Aggregated Injection strategy alone reduced velocity error by ~36% compared to a standard multi-input baseline.
- The model showed robust generalization across spatial locations and temporal phases, though pressure prediction remained more challenging for waveforms with sharp systolic peaks (FR2 group).
Computational Efficiency: Once trained, the model achieved a conservative inference speedup of approximately 36× compared to reference CFD simulations.
- CFD simulation of a single converged cardiac cycle (101 snapshots) took ~12 hours.
- M3PI-DeepONet inference for the same cycle took ~20 minutes.
- This speedup assumes the necessary branch-conditioning inputs (e.g., inlet waveform, initial state) are available.

Significance and Claims
The authors position this work as a significant step toward real-time, non-invasive clinical diagnostics for cardiovascular disease. By demonstrating that a physics-informed operator can learn the mapping between boundary/initial conditions and full spatio-temporal flow fields with high fidelity and low computational cost, the study validates the potential of DeepONets to replace or augment traditional CFD in clinical workflows.

The paper explicitly notes that the current study uses an idealized geometry and rigid walls, acknowledging that fluid-structure interaction (FSI) and patient-specific anatomies are not yet addressed. Furthermore, the results are framed as a measurement-conditioned assessment, as the model relies on specific branch inputs (like the initial state at $t=0$ ) that are currently derived from CFD in this benchmark. The authors claim that the primary contribution is not the elimination of CFD dependence entirely at this stage, but the demonstration that, given the required conditioning inputs, the full unsteady 3D hemodynamic field can be reconstructed rapidly from partial information. This establishes a critical prerequisite for future patient-specific, real-time hemodynamic modeling.

Technical Summary: Learning Unsteady Aneurysm Hemodynamics with Physics-Informed DeepONets

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