[Paper] Evolutionary Discovery of Sequence Acceleration Methods for Slab Geometry Neutron Transport

Source: arXiv - 2512.24559v1

Overview

A team of researchers from the neutron‑transport community has applied genetic programming—an evolutionary algorithm that “breeds” computer programs—to automatically invent new convergence‑acceleration formulas for discrete‑ordinates (S_N) neutron transport calculations in slab geometry. By letting the algorithm explore a vast space of mathematical expressions, they uncovered accelerators that outperform classic hand‑crafted methods such as Aitken’s Δ² and Wynn’s ε‑algorithm on a representative suite of transport problems.

Key Contributions

• Automated discovery pipeline: Designed a genetic‑programming framework that evolves symbolic acceleration formulas directly from raw iteration data.
• Novel accelerator: Identified a compact expression involving second‑order differences and cross‑product terms that improves convergence in >75 % of test cases—roughly twice the success rate of traditional accelerators.
• Benchmark suite: Created a diverse set of slab‑geometry neutron‑transport problems (varying scattering ratios, source terms, and angular discretizations) to rigorously evaluate both classic and evolved methods.
• Proof‑of‑concept for experimental mathematics: Demonstrated that evolutionary search can generate useful numerical algorithms, opening a pathway for similar discoveries in other computational physics domains.

Methodology

1. Problem formulation: The authors solved a series of one‑dimensional slab‑geometry neutron transport equations using the S_N discrete ordinates method, generating sequences of scalar flux iterates that converge slowly for many configurations.
2. Genetic programming (GP) setup:
   • Terminal set: Raw iteration values, first and second differences, and basic arithmetic operators.
   • Function set: +, –, ×, ÷, and a limited set of nonlinear functions (e.g., square, absolute value).
   • Fitness metric: Ratio of the number of iterations required to reach a prescribed tolerance with the candidate accelerator versus the unaccelerated sequence. Higher fitness = fewer extra iterations.
3. Evolutionary loop: Randomly generated candidate formulas (individuals) were evaluated on the benchmark suite, the best performers were selected, recombined (crossover), and mutated to produce the next generation. The process continued for several hundred generations until convergence of fitness.
4. Post‑processing: The top‑performing formulas were simplified analytically and tested on a held‑out set of problems to verify generalization.

Results & Findings

• The evolved accelerator (a concise expression using second differences and a cross‑product term) succeeded in improving convergence for 75 % of the test problems, compared to ≈40 % for Aitken’s Δ² and ≈35 % for Wynn’s ε.
• On problems where it succeeded, the new method reduced the required iteration count by 30–50 % on average, translating into noticeable runtime savings.
• The GP‑derived formulas exhibited robustness across a range of scattering ratios (from highly absorbing to highly scattering media) and angular quadratures (S_4 to S_16).
• Importantly, the discovered accelerator does not rely on assumptions about linear convergence; instead, it adapts to the actual curvature observed in the iteration sequence, explaining its broader applicability.

Practical Implications

• Faster transport simulations: For developers of deterministic neutron‑transport codes (e.g., MCNP‑DET, PARTISN, or custom S_N solvers), plugging in the new accelerator can cut wall‑clock time, especially in large‑scale reactor or shielding analyses where many slab‑type sub‑problems are solved repeatedly.
• Reduced need for hand‑tuning: Classical accelerators often require problem‑specific parameter choices. The GP‑derived formula works “out‑of‑the‑box,” lowering the expertise barrier for new users.
• Extensible framework: The same evolutionary pipeline can be repurposed for other geometries (cylindrical, spherical) or for different iterative solvers (e.g., Krylov methods, source‑iteration in thermal‑hydraulics).
• Open‑source potential: If the authors release the GP code and benchmark suite, the community could collaboratively evolve even more sophisticated accelerators, perhaps incorporating machine‑learning‑based surrogate models.

Limitations & Future Work

• Scope limited to slab geometry: The current study does not address multi‑dimensional transport problems, where spatial coupling may alter convergence behavior.
• Symbolic complexity ceiling: To keep formulas interpretable, the GP was constrained to relatively simple operators; richer function sets (e.g., trigonometric or exponential terms) might yield stronger accelerators but at the cost of readability.
• Computational cost of discovery: Evolving the formulas required substantial CPU time (hundreds of core‑hours). While this is a one‑off expense, scaling the approach to higher‑dimensional problems will demand more efficient search strategies.
• Robustness to noisy data: The method assumes clean iteration sequences; real‑world codes that introduce rounding errors or stochastic noise may affect the accelerator’s performance. Future work could integrate noise‑tolerant fitness measures.

Authors

• Japan K. Patel
• Barry D. Ganapol
• Anthony Magliari
• Matthew C. Schmidt
• Todd A. Wareing

Paper Information

• arXiv ID: 2512.24559v1
• Categories: cs.NE
• Published: December 31, 2025
• PDF: Download PDF