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1. Multimodal neural operators for real-time biomechanical modelling of traumatic brain injury

   https://doi.org/10.1016/j.cmpb.2026.109398  

   Agarwal, Anusha; Sarkar, Dibakar Roy; Goswami, Somdatta (August 2026, Computer Methods and Programs in Biomedicine)

   Background: Traumatic brain injury (TBI) remains a major public health concern, with over 69 million cases annually worldwide. Accurate patient-specific biomechanical modeling is critical for injury risk assessment, but it requires integrating heterogeneous data sources such as volumetric neuroimaging, scalar demographic parameters, and acquisition metadata. Conventional finite element solvers can perform this modeling, yet they remain too computationally expensive for time-sensitive clinical settings. Neural operators have emerged as a promising alternative by learning resolution-invariant mappings between function spaces at orders-of-magnitude faster inference. However, while recent work has introduced parameter-conditioned and multi-input operator architectures for incorporating auxiliary scalar or geometric inputs, the systematic integration of volumetric medical imaging with heterogeneous scalar metadata within an operator learning framework remains underexplored, particularly for biomechanical prediction tasks involving patient-specific anatomical and demographic variability. Objective: This study presents a systematic investigation of multimodal neural operator architectures for brain biomechanics, evaluating strategies for fusing heterogeneous input modalities such as volumetric anatomical imaging, scalar demographic features, and acquisition parameters to predict full-field brain displacement fields from MRE data. Methods: We reformulated TBI modeling as a multimodal operator learning problem and proposed two fusion strategies: field projection for Fourier Neural Operator (FNO) based architectures (broadcasting scalars onto spatial grids) and branch decomposition for Deep Operator Networks (DeepONet) (separate encoding with multiplicative fusion). Four architectures (FNO, Factorized FNO (F-FNO), Multi-Grid FNO (MG-FNO), DeepONet) were extended with multimodal fusion mechanisms and evaluated on 249 in vivo Magnetic Resonance Elastography (MRE) datasets across physiologically relevant frequencies (20 to 90 Hz). Results: DeepONet achieved the highest accuracy on real displacement fields (MSE = 0.0039, 90.0% accuracy) with the fastest inference (3.83 it/s) and fewest parameters (2.09M), while MG-FNO achieved the best performance on imaginary fields (MSE = 0.0058, 88.3% accuracy) with the lowest GPU memory among FNO variants (7.12 GB). No single architecture dominated across all criteria, revealing distinct trade-offs between accuracy, spatial fidelity, and computational cost. Conclusion: The results demonstrate that neural operators augmented with multimodal fusion mechanisms can accurately predict full-field brain displacement from heterogeneous biomedical inputs, with inference times orders of magnitude faster than finite element solvers. The systematic comparison of fusion strategies and architectures provides practical guidance for selecting operator learning approaches in biomedical settings where heterogeneous data integration is required. 

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2. Hydrodynamic performance estimation of undulating elastic plates using the Fourier neural operator

   https://doi.org/10.1063/5.0293701  

   Jawetz, Christopher L; Alexeev, Alexander (October 2025, Physics of Fluids)

   Fluid–structure interactions (FSI) underpin diverse biological and engineering systems, yet computational modeling of these interactions remains resource-intensive and constrained by high-dimensional parameter spaces. To address these limitations, we develop a data-driven, two-stage surrogate modeling framework leveraging Fourier neural operators (FNO) to accurately predict the hydrodynamic performance of biomimetic elastic propulsors across a broad range of design parameters. Our approach first maps geometric and physical parameters to the propulsor dynamic bending pattern and then uses this bending pattern to predict thrust, power, and efficiency. Trained using direct FSI simulation data, our surrogate model achieves three orders of magnitude speedup compared to the direct simulations while maintaining high accuracy. Crucially, the use of intermediate kinematic modeling significantly reduces the overall model size and enhances accuracy. Our FNO modeling framework enables rapid exploration of large design spaces, indicating strong potential for application in the design and optimization of FSI problems. 

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3. Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flows

   Cao, Shuhao; Brarda, Francesco; Li, Rui_Peng; Xi, Yuanzhe (April 2025, ICLR 2025)

   Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new learning framework to address these issues. A new spatiotemporal adaptation is proposed to generalize any Fourier Neural Operator (FNO) variant to learn maps between Bochner spaces, which can perform an arbitrary-length temporal super-resolution for the first time. To better exploit this capacity, a new paradigm is proposed to refine the commonly adopted end-to-end neural operator training and evaluations with the help from the wisdom from traditional numerical PDE theory and techniques. Specifically, in the learning problems for the turbulent flow modeled by the Navier-Stokes Equations (NSE), the proposed paradigm trains an FNO only for a few epochs. Then, only the newly proposed spatiotemporal spectral convolution layer is fine-tuned without the frequency truncation. The spectral fine-tuning loss function uses a negative Sobolev norm for the first time in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is exact thanks to the Parseval identity. Moreover, unlike the difficult nonconvex optimization problems in the end-to-end training, this fine-tuning loss is convex. Numerical experiments on commonly used NSE benchmarks demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers under certain conditions. The source code is publicly available at https://github.com/scaomath/torch-cfd. 

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4. Deep neural operator for learning transient response of interpenetrating phase composites subject to dynamic loading

   https://doi.org/10.1007/s00466-023-02343-6  

   Lu, Minglei; Mohammadi, Ali; Meng, Zhaoxu; Meng, Xuhui; Li, Gang; Li, Zhen (May 2023, Computational Mechanics)

   Additive manufacturing has been recognized as an industrial technological revolution for manufacturing, which allows fabrication of materials with complex three-dimensional (3D) structures directly from computer-aided design models. Using two or more constituent materials with different physical and mechanical properties, it becomes possible to construct interpenetrating phase composites (IPCs) with 3D interconnected structures to provide superior mechanical properties as compared to the conventional reinforced composites with discrete particles or fibers. The mechanical properties of IPCs, especially response to dynamic loading, highly depend on their 3D structures. In general, for each specified structural design, it could take hours or days to perform either finite element analysis (FEA) or experiments to test the mechanical response of IPCs to a given dynamic load. To accelerate the physics-based prediction of mechanical properties of IPCs for various structural designs, we employ a deep neural operator (DNO) to learn the transient response of IPCs under dynamic loading as surrogate of physics-based FEA models. We consider a 3D IPC beam formed by two metals with a ratio of Young’s modulus of 2.7, wherein random blocks of constituent materials are used to demonstrate the generality and robustness of the DNO model. To obtain FEA results of IPC properties, 5000 random time-dependent strain loads generated by a Gaussian process kennel are applied to the 3D IPC beam, and the reaction forces and stress fields inside the IPC beam under various loading are collected. Subsequently, the DNO model is trained using an incremental learning method with sequence-to-sequence training implemented in JAX, leading to a 100X speedup compared to widely used vanilla deep operator network models. After an offline training, the DNO model can act as surrogate of physics-based FEA to predict the transient mechanical response in terms of reaction force and stress distribution of the IPCs to various strain loads in one second at an accuracy of 98%. Also, the learned operator is able to provide extended prediction of the IPC beam subject to longer random strain loads at a reasonably well accuracy. Such superfast and accurate prediction of mechanical properties of IPCs could significantly accelerate the IPC structural design and related composite designs for desired mechanical properties. 

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5. HIOS: Hierarchical Inter-Operator Scheduler for Real-Time Inference of DAG-Structured Deep Learning Models on Multiple GPUs

   https://doi.org/10.1109/CLUSTER52292.2023.00016  

   Kundu, Turja; Shu, Tong (October 2023, IEEE)

   Neural-network-enabled data analysis in real-time scientific applications imposes stringent requirements on inference latency. Meanwhile, recent deep learning (DL) model design trends to replace a single branch with multiple branches for high prediction accuracy and robustness, which makes interoperator parallelization become an effective approach to improve inference latency. However, existing inter-operator parallelization techniques for inference acceleration are mainly focused on utilization optimization in a single GPU. With the data size of an input sample and the scale of a DL model ever-growing, the limited resource of a single GPU is insufficient to support the parallel execution of large operators. In order to break this limitation, we study hybrid inter-operator parallelism both among multiple GPUs and in each GPU. In this paper, we design and implement a hierarchical inter-operator scheduler (HIOS) to automatically distribute large operators onto different GPUs and group small operators in the same GPU for parallel execution. Particularly, we propose a novel scheduling algorithm, named HIOS-LP, which consists of inter-GPU operator parallelization through iterative longest-path (LP) mapping and intra-GPU operator parallelization based on a sliding window. In addition to extensive simulation results, experiments with modern convolutional neural network benchmarks demonstrate that our HIOS-LP outperforms the state-of-the-art inter-operator scheduling algorithm IOS by up to 17% in real systems. 

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