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

This paper demonstrates that for DeepONet and S-DeepONet architectures, single-branch designs outperform multi-branch variants in tightly coupled multiphysics systems by fostering shared latent representations, while multi-branch approaches remain superior for decoupled tasks, collectively enabling full-field predictions up to $1.8 \times 10^4$ times faster than traditional solvers.

Original authors: Jaewan Park, Kazuma Kobayashi, Qibang Liu, Seid Koric, Diab Abueidda, Syed Bahauddin Alam

Published 2026-06-12

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jaewan Park, Kazuma Kobayashi, Qibang Liu, Seid Koric, Diab Abueidda, Syed Bahauddin Alam

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a computer to predict how complex physical systems behave—like how heat spreads through a metal block while electricity flows through it, or how steel cools and changes shape as it solidifies. These aren't just simple problems; they are "multiphysics" puzzles where different forces (heat, electricity, stress) are tightly tangled together, influencing each other constantly.

For a long time, scientists have used powerful computer simulations (like Finite Element Analysis) to solve these puzzles. But these simulations are slow, like trying to solve a Rubik's cube by checking every single move one by one. To speed things up, researchers use "surrogate models"—AI shortcuts that learn from past simulations to predict the future instantly.

This paper asks a very practical question: When building these AI shortcuts, should we design them to treat different physical forces as separate teams working in parallel, or as one big, unified team working together?

The authors tested two main architectural styles using a type of AI called DeepONet:

The "Single-Branch" Team (The Unified Squad): Imagine a single group of workers who all look at the entire picture together. They share a common notebook (a shared "latent space") where they write down everything they see. They discuss the heat, the electricity, and the stress all at the same time, understanding how one affects the other immediately.
The "Multi-Branch" Team (The Specialized Departments): Imagine two separate teams. One team only looks at the heat, and the other team only looks at the electricity. They work in isolation, write their own reports, and then only at the very end do they slap their reports together to see the final result.

The Experiments: Three Different Puzzles

The researchers tested these two approaches on three different scenarios to see which "team structure" worked best.

1. The Simple Puzzle (Reaction-Diffusion)

The Scenario: A single physical process (like a chemical spreading in water) with two different inputs (a source and a reaction rate).
The Result: The Multi-Branch team won. Since the inputs were independent (one didn't really change the other), having specialized experts who focused on just one thing allowed them to be slightly more precise. It's like having a specialist for the engine and a specialist for the tires when building a car; they don't need to talk much to do their jobs well.

2. The Tangled Puzzle (Electro-Thermal)

The Scenario: Electricity flowing through a wire creates heat, and that heat changes how electricity flows. They are locked in a dance where you can't separate them.
The Result: The Single-Branch team crushed it. The Multi-Branch team struggled because by the time they tried to combine their separate reports, they had already missed the subtle ways the heat and electricity were influencing each other in real-time. The Unified Squad, sharing a common understanding from the start, captured these interactions perfectly. The error for the Multi-Branch team was huge (over 15% in some cases), while the Single-Branch team was nearly perfect (under 1%).

3. The Heavy Lifting Puzzle (Steel Solidification)

The Scenario: Molten steel cooling down. As it cools, it shrinks and creates stress. The temperature dictates the stress, and the stress changes the temperature. It's a highly complex, "path-dependent" system (the history of how it cooled matters).
The Result: Again, the Single-Branch team was the clear winner. The steel's behavior is so tightly coupled that separating the "temperature team" from the "stress team" led to significant errors. The Unified Squad, which learned the relationship between heat and stress simultaneously, predicted the stress levels much more accurately.

The Big Takeaway

The paper's main discovery is a simple rule of thumb for designing these AI models: Match the AI's structure to the physics.

If the physics are "decoupled" (independent): Use Multi-Branch networks. Let the specialists do their own thing.
If the physics are "coupled" (tightly tangled): Use Single-Branch networks. Force the AI to learn the connections between the forces from the very beginning, just like a monolithic (all-in-one) solver in traditional engineering.

The Speed Bonus

Beyond just accuracy, the paper highlights a massive speed advantage. Once these AI models are trained, they can predict the behavior of these complex systems 18,000 times faster than the traditional, heavy-duty computer simulations.

Traditional Solver: Takes about 5.5 minutes to solve one scenario.
AI Surrogate: Takes about 0.02 seconds.

This means that tasks that used to take hours or days can now be done in milliseconds, opening the door for real-time monitoring and control in things like manufacturing and energy systems.

Summary Analogy

Think of it like cooking a complex stew.

If you are making a salad where the tomatoes don't change the lettuce, you can have one person chop the tomatoes and another chop the lettuce, then mix them at the end (Multi-Branch).
But if you are making a stew where the meat changes the flavor of the broth, and the broth changes how the meat cooks, you need one chef who tastes, adjusts, and stirs everything together constantly (Single-Branch). If you try to cook the meat and the broth separately and mix them at the end, the stew will taste wrong.

This paper proves that for "stews" (tightly coupled physics), the single chef approach is not just better; it's necessary for getting the right result.

Technical Summary: Single vs. Multiple Branches in DeepONet and S-DeepONet

Problem Statement

Real-time prediction of complex physical systems requires surrogate models capable of learning from data while accurately representing strong multiphysics coupling. While Deep Operator Networks (DeepONets) have demonstrated success in single-physics problems, their effectiveness in capturing nonlinear interactions in tightly coupled systems (e.g., thermo-mechanical or electro-thermal coupling) remains underexplored. A critical gap exists in understanding whether the neural operator architecture should reflect the physical coupling strength of the system it aims to model. Specifically, it is unclear whether single-branch networks (which share latent representations) or multi-branch networks (which encode inputs independently and fuse them later) are superior for different coupling regimes.

Methodology

The authors conduct a systematic topological ablation study comparing single-branch and multi-branch (MIONet-style) designs within two operator families: the classical feedforward DeepONet and the sequential S-DeepONet (which utilizes Gated Recurrent Units, GRUs, to handle path-dependent inputs).

The study evaluates these architectures across three representative benchmark problems of increasing physical complexity:

Single-Physics Reaction-Diffusion: A transport-dominated process with heterogeneous sources and reaction coefficients. Inputs are spatial functions ( $u_0(x), k(x)$ ).
Coupled Electro-Thermal Diffusion: A system involving electrical conduction and thermal diffusion where conductivity is temperature-dependent and Joule heating occurs. Inputs are time-dependent sequences ( $Q_{ext}(t), \rho_e(t)$ ). Both coupled (mutually dependent fields) and uncoupled (independent fields) scenarios are tested.
Viscoplastic Thermo-Mechanical Steel Solidification: A complex continuous casting model involving heat transfer with latent heat, phase transformations (austenite/delta-ferrite), and viscoplastic deformation. Inputs are time-dependent heat flux and displacement histories. Both coupled and uncoupled (fixed temperature field) scenarios are analyzed.

Architectural Variants:

Single-Branch: All input functions are concatenated and encoded jointly into a shared latent space (using FNNs for spatial inputs or GRUs for sequential inputs).
Multi-Branch (MIONet-style): Each input function is encoded independently by its own branch network. The branch outputs are merged via an element-wise (Hadamard) product before interacting with the trunk network.

Performance is quantified using relative $L_2$ error for single-physics and coupled electro-thermal cases, and Mean Absolute Error (MAE) for the thermo-mechanical case to avoid instability near zero-stress regions. Robustness is also tested via additive Gaussian noise injection.

Key Results

The study reveals a distinct regime-dependent performance trend:

Tightly Coupled Systems: In strongly coupled multiphysics problems (electro-thermal and thermo-mechanical), single-branch networks consistently outperform multi-branch variants.
- Electro-Thermal: The single-branch S-DeepONet achieved significantly lower errors (e.g., ~0.76% vs. ~4.23% for electrical potential in the coupled regime). The multi-branch model exhibited long-tailed error distributions, particularly for electrical potential.
- Thermo-Mechanical: The single-branch model yielded a stress MAE of 0.12 MPa compared to 0.219 MPa for the two-branch model. It better preserved peak tensile values and stress reversals critical for failure prediction.
- Reasoning: Single-branch architectures encourage shared latent representations, allowing the network to learn cross-terms and nonlinear inter-field dependencies early in the encoding process, analogous to monolithic solvers in classical numerical analysis.
Decoupled or Single-Physics Systems: In single-physics or uncoupled multiphysics scenarios, multi-branch designs are either superior or perform comparably.
- In the reaction-diffusion benchmark (single physics), the multi-branch MIONet variant achieved lower error (2.55%) than the single-branch (3.40%).
- In uncoupled electro-thermal and thermo-mechanical tests, the multi-branch model showed slight accuracy advantages or equivalent performance, benefiting from modality-specific encoders for independent inputs.
Computational Efficiency:
- Training: Single-branch models generally required fewer trainable parameters and less training time (e.g., 1.4 hours vs. 2.8 hours for the thermo-mechanical case).
- Inference: Once trained, all models provided full-field predictions in milliseconds. The trained S-DeepONet models achieved speedups of approximately $1.8 \times 10^4$ times compared to high-fidelity finite element analysis (FEA) solvers running on HPC systems.
Robustness: Under noisy input conditions (3–15% additive Gaussian noise), the trend persisted: single-branch models remained more robust for coupled physics, while multi-branch models performed better for uncoupled cases.

Significance and Claims

The paper claims that the choice of neural operator architecture should be guided by the strength of physical coupling in the target system:

For Strongly Coupled Systems: Single-branch designs are the preferred architecture. They mimic the behavior of monolithic solvers by propagating multiphysics interactions through a shared global system, thereby capturing intrinsic cross-coupling terms more effectively than factorized (multi-branch) approaches.
For Decoupled Systems: Multi-branch designs remain advantageous due to their modularity and capacity to encode modality-specific priors without unnecessary parameter sharing.

The authors emphasize that these findings are empirical observations specific to DeepONet and S-DeepONet topologies. They do not claim universal applicability to other operator families (e.g., Fourier Neural Operators or attention-based architectures), noting that cross-family evaluation remains an open question.

Ultimately, the work demonstrates that aligning neural operator architecture with the underlying physics enables the creation of scalable, real-time surrogate models for complex multiphysics systems, overcoming the computational bottlenecks of traditional high-fidelity solvers.