Robust prediction of parameterized cardiovascular hemodynamics using deep operator networks with time normalization

Computational modeling of cardiovascular hemodynamics is essential for understanding disease mechanisms. While one-dimensional (1D) numerical solvers are widely used, their cumulative computational cost becomes prohibitive in many-query scenarios - such as global sensitivity analysis and Bayesian parameter estimation - that require thousands of iterative evaluations. This study presents a parameterized Deep Operator Network (DeepONet) framework integrated with time normalization to enable rapid and robust surrogate modeling.

The proposed framework employs a two-step time normalization strategy comprising cycle and period normalization. A parameterized trunk network explicitly encodes heart rate, arterial stiffness, and relative arterial length to compensate for temporal scaling and facilitate explicit parameterization. The method was evaluated on simulated 1D cardiovascular cases using a testing protocol that included out-of-distribution scenarios with parameters extending significantly beyond the training range.

The model achieved high accuracy within the training distribution, with a root mean squared error (RMSE) below 0.5 mmHg. In out-of-distribution regimes where standard models showed performance degradation (RMSE >20 mmHg), the proposed framework maintained relatively robust performance (mean RMSE <7 mmHg). Inference time was approximately 0.00038 s per waveform, representing a computational acceleration of around 150,000 times compared to numerical solvers. The framework's generalizability was further corroborated on a periodic structural dynamics problem predicting cantilever beam displacements.

These results demonstrate that integrating time normalization with parameterized operator learning enables robust modeling of periodic physiological signals. This approach provides a computational foundation for solving computationally intensive inverse problems and performing large-scale sensitivity analyses, where standard numerical simulations are practically infeasible.