Surrogate Modeling for Neutron Transport: A Neural Operator Approach

This paper demonstrates that DeepONet and Fourier Neural Operator (FNO) architectures serve as highly efficient and accurate surrogate models for neutron transport, achieving significant computational speedups over conventional solvers while maintaining precision across various scattering regimes and eigenvalue problems.

The Gist

Imagine you are trying to predict how a crowd of people will move through a complex building. In the world of nuclear reactors, these "people" are neutrons, and the "building" is the reactor core. To keep a reactor safe and efficient, engineers need to know exactly where every neutron is going, how fast it's moving, and what it hits along the way.

This is a job for the Neutron Transport Equation (NTE). It's a incredibly complex math problem that describes the chaotic dance of neutrons.

The Problem: The "Slow Motion" Camera

Traditionally, to solve this math problem, scientists use powerful computers to simulate every single step of every neutron. It's like trying to film a high-speed race in slow motion, frame by frame, to understand the physics.

• The Good News: It's incredibly accurate.
• The Bad News: It takes a long time. If you want to design a new reactor or test a thousand different safety scenarios, waiting hours or days for each calculation is impossible. You need a "real-time" answer.

The Solution: The "Magic Predictor" (Neural Operators)

This paper introduces a new way to solve the problem using Artificial Intelligence (AI), specifically something called Neural Operators.

Think of traditional AI (like the kind that recognizes cats in photos) as a student who memorizes specific answers. If you show it a cat, it says "cat." If you show it a slightly different cat, it might get confused.

Neural Operators are different. They are like a master chef who learns the recipe of cooking, not just specific dishes.

• Instead of memorizing one specific neutron path, the AI learns the rules of the game.
• Once it learns the rules, you can give it any new scenario (a new reactor shape, a new fuel type, a new source of neutrons), and it can instantly predict the outcome without having to "re-simulate" the whole race from scratch.

The Two Contenders: DeepONet vs. FNO

The researchers tested two different "chefs" (AI architectures) to see who could predict the neutron dance best:

1. DeepONet (The Speedster):

   • Analogy: Imagine a sprinter. It's incredibly fast at giving you an answer.
   • Performance: It was the fastest model, often taking less than 1% of the time a traditional computer needs. However, it was slightly less precise, like a sprinter who finishes the race a few inches off the perfect line.

2. FNO (The Precision Architect):

   • Analogy: Imagine a master architect who uses a special blueprint (Fourier transforms) to see the whole building at once.
   • Performance: It was slightly slower than DeepONet but gave a more accurate prediction, hitting the "perfect line" more often.

The Results: A Game Changer

The researchers put these AI models to the test in two scenarios:

1. The Fixed Source Test (The "What If" Scenario)
They asked: "If we put a neutron source here, what happens?"

• Result: Both AI models were hundreds of times faster than the traditional method.
• Accuracy: They were accurate enough to be trusted. The "Precision Architect" (FNO) was slightly more accurate, but the "Speedster" (DeepONet) was fast enough to be useful for real-time decisions.
• Generalization: Even when they tested the AI with source patterns it had never seen before, it still worked well. It truly learned the physics, not just the data.

2. The Eigenvalue Test (The "Criticality" Scenario)
This is the most important test for reactors: "Is the reactor critical? Will it sustain a chain reaction?"

• The Old Way: The computer has to run a loop, checking the answer, adjusting, and checking again, thousands of times. It's like trying to tune a radio by turning the knob back and forth very slowly.
• The New Way: The AI skips the loop. It looks at the current state and instantly predicts the next step.
• Result: The AI models reduced the calculation time to less than 0.1% of the original time. That's like turning a 10-hour calculation into a 30-second one! The predicted "criticality" numbers were very close to the real answer (within a tiny margin of error).

Why Does This Matter?

Imagine you are designing a new car.

• Before: You build a physical prototype, crash it, fix it, and build another. This takes months.
• With this AI: You have a "Digital Twin." You can simulate a million crashes in an afternoon, tweak the design instantly, and find the perfect car before you ever build a physical one.

This paper shows that Neural Operators can do this for nuclear reactors. They allow engineers to:

• Design safer reactors faster.
• Optimize fuel usage in real-time.
• Create "Digital Twins" of power plants that can predict problems before they happen.

The Bottom Line

The researchers have built a "shortcut" through the complex math of nuclear physics. It's not a cheat code; it's a smarter way of thinking. By teaching the computer the rules of neutron behavior rather than forcing it to calculate every step, they have unlocked the ability to solve nuclear problems in real-time, opening the door to a new era of nuclear safety and design.

Technical Summary

1. Problem Statement

The neutron transport equation (NTE) is fundamental to reactor design, criticality safety, and shielding analysis. While deterministic solvers (like $S_N$) and Monte Carlo methods provide high-fidelity results, they are computationally expensive. This cost becomes a bottleneck for applications requiring rapid, repeated evaluations, such as:

• Design optimization and uncertainty quantification.
• Real-time digital twin frameworks for operational decision-making.
• Multiphysics coupling (e.g., neutronics with thermal-hydraulics).

Traditional machine learning approaches (e.g., standard neural networks or PINNs) often struggle with generalization; they typically learn discrete point-wise relationships for fixed physical configurations, requiring retraining when input parameters or source distributions change. The authors aim to overcome these limitations by developing neural operator surrogates that learn the continuous mapping between input functions (sources) and output functions (fluxes), enabling generalization across unseen scenarios without retraining.

2. Methodology

A. Neural Operator Frameworks

The study employs two state-of-the-art neural operator architectures to learn the solution operator for the NTE:

1. Deep Operator Network (DeepONet): Based on the Universal Approximation Theorem for operators. It consists of a BranchNet (encoding the input source function $Q(x, \mu)$) and a TrunkNet (encoding the query coordinates $(x, \mu)$). The output is the inner product of these two networks, allowing for function-to-function mapping.
2. Fourier Neural Operator (FNO): A data-driven approach that operates in the Fourier domain. It lifts the input to a latent space, applies learnable linear transformations in the frequency domain via Fast Fourier Transforms (FFT), and projects back to the spatial domain. A key feature is its ability to perform zero-shot super-resolution, meaning models trained on coarse grids can infer solutions on finer grids without retraining.

B. Problem Formulation

The study focuses on a one-dimensional slab geometry (10 cm length) with vacuum boundaries. Two problem types are addressed:

1. Fixed Source Problem: Solving for the angular flux $\psi(x, \mu)$ given an anisotropic source $Q(x, \mu)$. The cross-sections are fixed, but the source varies.
2. k-Eigenvalue Problem: Solving for the criticality eigenvalue ($k$) and flux distribution where neutrons are generated via fission. The neural operators replace the computationally intensive inner transport sweep loop in the power iteration method.

C. Training and Data Generation

• Data Source: High-fidelity $S_N$ solutions generated using a cell-averaged finite difference scheme with Gauss-Legendre quadrature ($N=16$ angles).
• Source Generation: Anisotropic sources were generated using Gaussian Random Fields (GRFs) with varying correlation lengths ($l_S$ for spatial, $l_Q$ for angular).
• Scattering Regimes: Three distinct models were trained for each architecture to cover different physical regimes defined by the scattering ratio ($c = \Sigma_s / \Sigma_t$):
  • $c = 0.1$: Absorption-dominated.
  • $c = 0.5$: Moderate scattering.
  • $c = 1.0$: Purely scattering (scattering-dominated).
• Normalization: Sources were normalized to ensure a unit total neutron emission.

3. Key Contributions

• Extension to Anisotropic Sources: Unlike previous work by the authors that focused on isotropic sources, this study extends neural operators to map anisotropic sources $Q(x, \mu)$ to angular fluxes $\psi(x, \mu)$, a more complex and realistic scenario.
• Generalization Across Regimes: The models demonstrate robust generalization to unseen source configurations (varying correlation lengths) and different scattering regimes without retraining.
• Integration into Eigenvalue Solvers: The authors successfully integrated these surrogates into the $S_N$ k-eigenvalue solver, replacing the iterative transport sweep with a single forward pass, drastically accelerating criticality calculations.
• Zero-Shot Super-Resolution: The FNO architecture was shown to maintain accuracy when applied to finer spatial/angular grids than those used during training.

4. Results

A. Fixed Source Performance

• Accuracy:
  • FNO consistently achieved higher predictive accuracy, with Mean Squared Error (MSE) for angular flux typically in the range of $10^{-4}$ to $10^{-5}$ and Mean Relative Error (MRE) for scalar flux $< 0.2\%$ (even in scattering-dominated regimes).
  • DeepONet showed slightly higher errors (MRE up to $\sim 1.6\%$ in absorption-dominated cases) but remained highly accurate.
• Generalization: Both models maintained low MSE across a wide parameter space of source correlation lengths, including "out-of-distribution" cases where test correlation lengths differed significantly from training data.
• Efficiency:
  • FNO required < 13% of the runtime of the traditional $S_N$ solver (down to 0.11% in scattering-dominated cases).
  • DeepONet was even faster, requiring < 7% of the runtime (down to 0.21% in scattering-dominated cases).
  • Speedups became more pronounced as the scattering ratio ($c$) increased.

B. Eigenvalue Problem Performance

• Accuracy: When integrated into the k-eigenvalue solver:
  • FNO showed deviations from reference eigenvalues ranging from -34 to +112 pcm (depending on mesh and cross-section).
  • DeepONet showed larger deviations, ranging from -92 to -135 pcm.
  • Both models preserved the correct ordering of eigenvalues across different fission cross-sections.
• Efficiency: On fine grids ($J=1000, N=32$), both neural operator solvers reduced runtime to < 0.1% of the traditional $S_N$ solver.
• Mesh Independence: The models demonstrated the ability to generalize to finer meshes ($J=1000$) than the training mesh ($J=100$), validating the zero-shot super-resolution capability of FNO and the robustness of DeepONet.

5. Significance and Conclusion

This work establishes neural operators as a viable, high-performance alternative to traditional transport solvers for nuclear engineering applications.

• Real-Time Capability: The massive speedups (orders of magnitude) enable real-time or near-real-time predictions, crucial for digital twins and online monitoring.
• Design Optimization: The ability to perform rapid, repeated evaluations makes these surrogates ideal for optimization studies and uncertainty quantification, which are often prohibitive with standard solvers.
• Multiphysics Potential: The efficiency of these models facilitates the tight coupling of neutronics with other physics (e.g., thermal-hydraulics), which is essential for advanced reactor simulation.
• Future Outlook: The authors suggest extending this framework to higher-dimensional geometries and developing models that can generalize across varying cross-section values, further bridging the gap between AI surrogates and practical reactor physics.

In summary, the paper demonstrates that FNO offers superior accuracy while DeepONet offers superior computational speed, and both provide a generalizable, efficient framework for solving complex neutron transport problems.