# Single vs. Multiple Branches in DeepONet and S-DeepONet: Network Architecture Follows Coupling in Multiphysics Systems

arXiv: https://arxiv.org/abs/2507.03660 · HTML: https://mlsanity.com/paper/2507.03660/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 38 of its 90 references that are in the map.
- First posted on arXiv 4 Jul 2025.

## Builds on (most cited first, 30 of 38)

- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Characterizing possible failure modes in physics-informed neural networks](https://mlsanity.com/paper/2109.01050/) (2021)
- [When and why PINNs fail to train: A neural tangent kernel perspective](https://mlsanity.com/paper/2007.14527/) (2020)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Pretraining Codomain Attention Neural Operators for Solving Multiphysics PDEs](https://mlsanity.com/paper/2403.12553/) (2024)
- [Multifidelity Deep Operator Networks For Data-Driven and Physics-Informed Problems](https://mlsanity.com/paper/2204.09157/) (2022)
- [Inverse-design of nonlinear mechanical metamaterials via video denoising diffusion models](https://mlsanity.com/paper/2305.19836/) (2023)
- [Sequential Deep Operator Networks (S-DeepONet) for Predicting Full-field Solutions Under Time-dependent Loads](https://mlsanity.com/paper/2306.08218/) (2023)
- [Geom-DeepONet: A Point-cloud-based Deep Operator Network for Field Predictions on 3D Parameterized Geometries](https://mlsanity.com/paper/2403.14788/) (2024)
- [Fourier-MIONet: Fourier-enhanced multiple-input neural operators for multiphase modeling of geological carbon sequestration](https://mlsanity.com/paper/2303.04778/) (2023)
- [Mitigating spectral bias for the multiscale operator learning](https://mlsanity.com/paper/2210.10890/) (2022)
- [On the training and generalization of deep operator networks](https://mlsanity.com/paper/2309.01020/) (2023)
- [ViTO: Vision Transformer-Operator](https://mlsanity.com/paper/2303.08891/) (2023)
- [The mixed deep energy method for resolving concentration features in finite strain hyperelasticity](https://mlsanity.com/paper/2104.09623/) (2021)
- [Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutions](https://mlsanity.com/paper/2308.07523/) (2023)
- [Explainable, Interpretable & Trustworthy AI for Intelligent Digital Twin: Case Study on Remaining Useful Life](https://mlsanity.com/paper/2301.06676/) (2023)
- [Improved generalization with deep neural operators for engineering systems: Path towards digital twin](https://mlsanity.com/paper/2301.06701/) (2023)
- [D2NO: Efficient Handling of Heterogeneous Input Function Spaces with Distributed Deep Neural Operators](https://mlsanity.com/paper/2310.18888/) (2023)
- [Diffeomorphism Neural Operator for various domains and parameters of partial differential equations](https://mlsanity.com/paper/2402.12475/) (2024)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [A deep learning energy-based method for classical elastoplasticity](https://mlsanity.com/paper/2209.06467/) (2022)
- [Advanced Deep Operator Networks to Predict Multiphysics Solution Fields in Materials Processing and Additive Manufacturing](https://mlsanity.com/paper/2403.14795/) (2024)
- [Virtual Sensing-Enabled Digital Twin Framework for Real-Time Monitoring of Nuclear Systems Leveraging Deep Neural Operators](https://mlsanity.com/paper/2410.13762/) (2024)

## Built on by (most cited first, 2 of 2)

- [Real-Time Sensing of Inaccessible Physical Fields via an Edge-Deployable Hardware-Portable Graph Neural Operator](https://mlsanity.com/paper/2604.01802/) (2026)
- [GenCP: Towards Generative Modeling Paradigm of Coupled Physics](https://mlsanity.com/paper/2601.19541/) (2026)

## Abstract (arXiv)

`Real-time prediction of complex physical systems requires surrogate models that learn from data while representing strong multiphysics coupling. Deep Operator Networks have shown success in single-physics problems, yet their effectiveness in capturing nonlinear interactions in coupled systems (such as thermo-mechanical or electro-thermal coupling) remains underexplored. Here we pose a practical question: should the architecture of a neural operator reflect the strength of physical coupling it aims to model? We compare single-branch and multi-branch designs, in both feedforward and sequential recurrent forms, across three representative systems: a reaction--diffusion problem with heterogeneous sources, a nonlinear thermo-electrical problem with temperature-dependent conductivity and Joule heating, and a viscoplastic thermo-mechanical model of steel solidification. Single-branch networks consistently outperform multi-branch variants in tightly coupled regimes by encouraging shared latent representations, whereas multi-branch designs remain favorable for decoupled or single-physics tasks. Once trained, these surrogates deliver full-field predictions up to 1.8 × 104 times faster than physics-based solvers.