# Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads

arXiv: https://arxiv.org/abs/2306.08218 · HTML: https://mlsanity.com/paper/2306.08218/

## Key facts

- Cited by 47 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Jun 2023.

## Built on by (most cited first, 30 of 47)

- [Basis-to-Basis Operator Learning Using Function Encoders](https://mlsanity.com/paper/2410.00171/) (2024)
- [Separable DeepONet: Breaking the Curse of Dimensionality in Physics-Informed Machine Learning](https://mlsanity.com/paper/2407.15887/) (2024)
- [TI-DeepONet: Learnable Time Integration for Stable Long-Term Extrapolation](https://mlsanity.com/paper/2505.17341/) (2025)
- [Virtual Sensing-Enabled Digital Twin Framework for Real-Time Monitoring of Nuclear Systems Leveraging Deep Neural Operators](https://mlsanity.com/paper/2410.13762/) (2024)
- [A finite operator learning technique for mapping the elastic properties of microstructures to their mechanical deformations](https://mlsanity.com/paper/2404.00074/) (2024)
- [Learning the Hodgkin-Huxley Model with Operator Learning Techniques](https://mlsanity.com/paper/2406.02173/) (2024)
- [Point-DeepONet: Predicting Nonlinear Fields on Non-Parametric Geometries under Variable Load Conditions](https://mlsanity.com/paper/2412.18362/) (2024)
- [Physics-Informed Latent Neural Operator for Real-time Predictions of time-dependent parametric PDEs](https://mlsanity.com/paper/2501.08428/) (2025)
- [AMR-Transformer: Enabling Efficient Long-range Interaction for Complex Neural Fluid Simulation](https://mlsanity.com/paper/2503.10257/) (2025)
- [Geometry-Informed Neural Operator Transformer](https://mlsanity.com/paper/2504.19452/) (2025)
- [Temporal Neural Operator for Modeling Time-Dependent Physical Phenomena](https://mlsanity.com/paper/2504.20249/) (2025)
- [On the locality of local neural operator in learning fluid dynamics](https://mlsanity.com/paper/2312.09820/) (2023)
- [Efficient Training of Deep Neural Operator Networks via Randomized Sampling](https://mlsanity.com/paper/2409.13280/) (2024)
- [A physics-informed transformer neural operator for learning generalized solutions of initial boundary value problems](https://mlsanity.com/paper/2412.09009/) (2024)
- [The Seismic Wavefield Common Task Framework](https://mlsanity.com/paper/2512.19927/) (2025)
- [Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains](https://mlsanity.com/paper/2410.04655/) (2024)
- [History-Aware Neural Operator: Robust Data-Driven Constitutive Modeling of Path-Dependent Materials](https://mlsanity.com/paper/2506.10352/) (2025)
- [Physics-Informed Time-Integrated DeepONet: Temporal Tangent Space Operator Learning for High-Accuracy Inference](https://mlsanity.com/paper/2508.05190/) (2025)
- [Liquid Fourier Latent Dynamics Networks for fast GPU-based numerical simulations in computational cardiology](https://mlsanity.com/paper/2408.09818/) (2024)
- [Enhanced Vascular Flow Simulations in Aortic Aneurysm via Physics-Informed Neural Networks and Deep Operator Networks](https://mlsanity.com/paper/2503.17402/) (2025)
- [Distribution-Free Uncertainty-Aware Virtual Sensing via Conformalized Neural Operators](https://mlsanity.com/paper/2507.11574/) (2025)
- [Probabilistic Predictions of Process-Induced Deformation in Carbon/Epoxy Composites Using a Deep Operator Network](https://mlsanity.com/paper/2512.13746/) (2025)
- [A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systems](https://mlsanity.com/paper/2603.14863/) (2026)
- [Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems](https://mlsanity.com/paper/2507.03660/) (2025)
- [Deep Neural Networks Inspired by Differential Equations](https://mlsanity.com/paper/2510.09685/) (2025)
- [Physics-informed DeepONet with stiffness-based loss functions for structural response prediction](https://mlsanity.com/paper/2409.00994/) (2024)
- [A Comparison of Parametric Dynamic Mode Decomposition Algorithms for Thermal-Hydraulics Applications](https://mlsanity.com/paper/2503.24205/) (2025)
- [MLPs and KANs for data-driven learning in physical problems: A performance comparison](https://mlsanity.com/paper/2504.11397/) (2025)
- [Physics-informed Multiple-Input Operators for efficient dynamic response prediction of structures](https://mlsanity.com/paper/2505.07090/) (2025)
- [Surrogate Modeling for Neutron Transport: A Neural Operator Approach](https://mlsanity.com/paper/2602.15890/) (2026)

## Abstract (arXiv)

Deep Operator Network (DeepONet), a recently introduced deep learning operator network, approximates linear and nonlinear solution operators by taking parametric functions (infinite-dimensional objects) as inputs and mapping them to solution functions in contrast to classical neural networks that need re-training for every new set of parametric inputs. In this work, we have extended the classical formulation of DeepONets by introducing sequential learning models like the gated recurrent unit (GRU) and long short-term memory (LSTM) in the branch network to allow for accurate predictions of the solution contour plots under parametric and time-dependent loading histories. Two example problems, one on transient heat transfer and the other on path-dependent plastic loading, were shown to demonstrate the capabilities of the new architectures compared to the benchmark DeepONet model with a feed-forward neural network (FNN) in the branch. Despite being more computationally expensive, the GRU- and LSTM-DeepONets lowered the prediction error by half (0.06% vs. 0.12%) compared to FNN-DeepONet in the heat transfer problem, and by 2.5 times (0.85% vs. 3%) in the plasticity problem. In all cases, the proposed DeepONets achieved a prediction R2 value of above 0.995, indicating superior accuracy. Results show that once trained, the proposed DeepONets can accurately predict the final full-field solution over the entire domain and are at least two orders of magnitude faster than direct finite element simulations, rendering it an accurate and robust surrogate model for rapid preliminary evaluations.