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Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads
Junyan He, Shashank Kushwaha, Jaewan Park, Seid Koric, Diab Abueidda, Iwona Jasiuk
TL;DR
Time-dependent loading calls for models that retain input-history causality while predicting full-field solutions, a gap for feed-forward DeepONets and standard recurrent networks. This work places GRU or LSTM sequence learning in the DeepONet branch; in heat-transfer and plasticity examples, the recurrent variants improve prediction accuracy over the FNN-DeepONet benchmark.
Problem
FNN-branch DeepONets do not retain causal relationships in time-dependent inputs, while standard recurrent networks are not designed to predict full-field spatial contours.
Source: §1 · pp. 2–3Method
The proposed approach places GRU or LSTM sequence-learning models in the DeepONet branch to encode time-dependent loading histories for full-field prediction.
Source: §1 · p. 3, §4 · p. 17Results
Across the heat-transfer and plasticity examples, GRU- and LSTM-based DeepONets improve prediction accuracy over FNN-DeepONet, with GRU highest in the plasticity study.
Source: §3.1 · p. 12, §3.2 · p. 14Takeaways & Limitations
The trained sequential DeepONets are presented as accurate, efficient surrogates for FE simulations in applications requiring many forward evaluations.
Source: §4 · p. 18The models predict only the end state of a time-dependent loading history, not its full evolution.
Source: §4 · p. 17
Abstract
from arXiv · showhide
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 $R^2$ 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.
1. Introduction
DeepONet addresses the need to avoid retraining for changing parametric inputs, but its feed-forward branch does not retain causal relationships in time-dependent inputs. This work combines recurrent sequence models with DeepONet to predict full-field solutions under time-dependent loads.
1. Introduction
- Deep-learning surrogates and physics-informed approaches have been applied to diverse physical problems, but changing loads, boundary conditions, material properties, or geometry can require retraining.DeepONet was introduced to map parametric functions to solution spaces without retraining for every input set.Source: §1 · p. 2
- Classical DeepONets map input functions to solution functions, but their feed-forward branch does not retain causal relationships in time-dependent inputs.DeepONet uses a branch network to encode input functions and a trunk network to encode domain geometry.Source: §1 · p. 2
- Recurrent networks such as GRU and LSTM encode time-dependent signals, but standard versions generally predict one-dimensional output sequences rather than spatial full-field contours.ConvLSTM combines temporal LSTM encoding with convolutional spatial encoding, but is limited to structured grids.Source: §1 · pp. 2–3
- The work combines GRU and LSTM sequence learning in the DeepONet branch to predict full-field solutions for transient heat transfer and path-dependent plasticity.The authors describe this combination as a first in the literature.Source: §1 · p. 3
- Time-dependent nonlinear analyses often require capturing load histories, while repeated finite-element evaluations can be costly in sensitivity analysis, uncertainty quantification, and optimization.The paper motivates accurate surrogate models that predict full-field quantities of interest.Source: §1 · p. 3
2. Methods
The methods use DeepONet to map time-dependent loads and spatial coordinates to final temperature or stress fields, replacing the feed-forward branch with GRU or LSTM alternatives and evaluating both on finite-element-generated datasets.
2.1.1. FNN-DeepONet
- DeepONet encodes load magnitudes in its branch and nodal coordinates in its trunk, combining their outputs to predict the final field over the domain.The FNN baseline uses 101 time steps for the load vector and an N × 2 coordinate matrix for the domain nodes.Source: §2.1.1 · pp. 4–5
2.1.2. Sequential DeepONets
- GRU- and LSTM-DeepONets replace the FNN branch with sequential networks while retaining the baseline branch's input and output signatures.Both variants use the same inputs, outputs, optimizer, loss function, and training epochs as the FNN baseline.Source: §2.1.2 · pp. 5–6
- The GRU branch uses a four-layer encoder-decoder with 256- and 128-unit encoding layers, followed by 128- and 256-unit decoding layers.A time-distributed dense layer produces a hidden dimension of 101; the resulting GRU-DeepONet has 792422 trainable parameters.Source: §2.1.2 · p. 6
- The LSTM branch likewise uses four encoder-decoder layers, with 256- and 128-unit encoder layers and 128- and 256-unit decoder layers.Its trunk matches the GRU trunk, and the LSTM-DeepONet has 1039206 trainable parameters.Source: §2.1.2 · p. 6
2.2. Data generation
- The study compares the two sequential DeepONets with the FNN-DeepONet in transient heat transfer and structural deformation.The two application problems are introduced as the settings for the model comparison.Source: §2.2 · p. 6
2.2.1. Transient heat transfer
- The heat-transfer case models nonlinear solidification in a 2D caster slice under a time-dependent boundary heat flux, with other surfaces insulated.The 4000 finite-element simulations use six-point heat-flux histories over 17 seconds, with randomly sampled control points and flux values.Source: §2.2.1 · pp. 6–8
2.2.2. History-dependent plastic deformation
- The plasticity case applies a time-dependent displacement to a dogbone specimen modeled with J2 plasticity and linear isotropic hardening.Six control points define each loading path, nominal axial strain stays below 5%, and 15000 finite-element simulations provide von Mises stress labels.Source: §2.2.2 · pp. 8–9
3. Results and discussion
Across transient heat transfer and history-dependent plasticity, GRU- and LSTM-DeepONets outperform the FNN-based model, though plasticity predictions include substantial outliers. Sequential models also require longer training, while inference is much faster than finite-element simulation.
3.1. Transient heat transfer
- By half, GRU- and LSTM-DeepONets lowered average L2 error versus FNN-DeepONet in heat transfer; all three achieved over 1500 times FE speedup.The recurrent models required 1.5 and 1.7 times longer training, respectively.Source: Table 6 · p. 11, §3.1 · p. 12
- Below 0.2% error was achieved with 2,000 heat-transfer training points; across training fractions, errors stayed below 0.2% and R2 exceeded 0.999.The LSTM model had the highest accuracy, though its difference from GRU was minimal and five-fold results showed minimal variation.Source: §3.1 · pp. 10–12
3.2. History-dependent plastic deformation
- 2.5 times more accurate than FNN-DeepONet, GRU- and LSTM-DeepONets required 1.5 times longer training, while all models achieved at least 200 times FE speedup in plasticity.Plasticity worst-case errors reached 615%, and at least 3.4% of cases exceeded 5% error.Source: Table 9 · p. 14, §3.2 · p. 14
- With 80% of data used for training, plasticity accuracy was best across all models; GRU cross-validation was consistent, and GRU and LSTM did not overfit.The GRU model trained 14.2% faster than LSTM.Source: §3.2 · pp. 13–14
- At the 99th error percentile in plasticity, GRU and LSTM captured stress-concentration locations relatively accurately, while FNN missed both; the three models performed well through the 90th percentile.The worst-case contours for all three models missed the stress-concentration locations and magnitudes.Source: §3.2 · p. 15, Figure 13 · p. 15, Figure 14 · p. 16
- As mean ground-truth stress increased, prediction error generally decreased; mean-squared-error training gives low-stress cases less influence, so they improve less efficiently.The authors propose investigating relative loss in future work.Source: §3.2 · pp. 16–17, Figure 15 · p. 16
4. Conclusions, limitations, and future work
Sequential GRU- and LSTM-DeepONets improve full-field predictions over the FNN-DeepONet benchmark for time-dependent loads, with especially strong gains in plasticity. The conclusions also identify current limits in prediction scope and data sampling, while describing practical uses and future extensions.
4. Conclusions, limitations, and future work
- GRU/LSTM-DeepONets halved average test error versus FNN-DeepONet and reduced plasticity prediction error 2.5 times, predicting Mises stress accurately up to the 90th percentile of prediction error.The authors recommend GRU-DeepONet over both FNN-DeepONet and LSTM-DeepONet; it has fewer trainable parameters, similar accuracy, and faster training.Source: §4 · p. 17
- Once trained, sequential DeepONets infer accurate full-field results at least two orders of magnitude faster than classical nonlinear numerical methods.The authors describe use as an FE surrogate in high-fidelity multiscale modeling, optimization, and design when many forward evaluations are needed.Source: §4 · pp. 17–18
- Current models predict only the end state, and uniform sampling does not account for real-world load imbalance around design loads and rare extremes.The authors identify resampling treatments as a future approach to account for training-data imbalance.Source: §4 · p. 17
- Future work will extend the architectures to three-dimensional domains and complete time histories; trained weights can also be deployed on laptops or edge devices for inference.The authors also identify online control scenarios as a possible use.Source: §4 · p. 18
Replication of results
The study's supporting data and source code are available through the research group's GitHub repository, with availability upon request during review.
Replication of results
- The study's data and source code are available through the Jasiuk Research Group GitHub repository, and can be requested from the corresponding author during review.The passage states the link would be made public upon publication.Source: Replication of results · p. 18
CRediT author contributions
The listed contributors cover methodology, analysis, investigation, supervision, resources, writing, and funding acquisition, with responsibilities distributed across the authors.
CRediT author contributions
- He and Kushwaha contributed methodology, formal analysis, investigation, and original-draft writing.Both are listed with the same contribution roles.Source: CRediT author contributions · p. 18
- Park contributed methodology, investigation, and original-draft writing; Koric contributed conceptualization, methodology, supervision, resources, original-draft writing, and funding acquisition.Source: CRediT author contributions · p. 18
- Abueidda and Jasiuk contributed supervision and review-and-editing of the manuscript.Source: CRediT author contributions · p. 18