Scientific discovery increasingly involves more than applying established mathematical equations. In computational physics, researchers are exploring whether computers can help discover new mathematical relationships, particularly when conventional numerical methods struggle to deliver accurate results efficiently. A recent study demonstrates how evolutionary computing can contribute to this effort by developing a mathematical formula specifically designed to improve the convergence of neutron transport calculations.
Published in Nuclear Science and Engineering, American Nuclear Society’s flagship journal, the research article, “Evolutionary Discovery of Sequence Acceleration Methods for Slab Geometry Neutron Transport”, was authored by Japan K. Patel, Barry D. Ganapol, Anthony Magliari, Matthew C. Schmidt and Todd A. Wareing. The paper lists the authors’ affiliations as Gateway Scripts, University of Arizona, Varian Medical Systems, and Washington University in St Louis.
The researchers applied genetic programming, an evolutionary computing technique, to discover a mathematical expression capable of accelerating the convergence of discrete ordinates solutions to neutron transport problems. The resulting formula improved convergence in approximately 78% of the test cases, compared with 39% for Aitken’s delta-squared method and 32% for the Wynn-epsilon algorithm. These findings suggest that data-driven mathematical discovery could complement conventional numerical analysis in computational physics.
Why neutron transport calculations matter
Neutrons are subatomic particles that interact with matter through scattering, absorption and other nuclear processes. Predicting how they travel through a material is an important problem in nuclear engineering and radiation transport. Mathematical models of neutron behaviour contribute to calculations relevant to nuclear reactor analysis, radiation shielding and medical physics.
Such calculations require numerical methods capable of representing particle movement and interactions with sufficient precision. In many applications, the challenge is not simply to obtain a numerical answer, but to establish that the answer has converged towards a reliable solution. A calculation that converges slowly may require substantial computational effort, particularly when high precision is needed.
The study focuses on neutron transport in slab geometry, a simplified configuration in which material is represented as a one-dimensional slab. Although this geometry does not capture every feature of a real reactor, it provides a useful mathematical setting for investigating transport equations and evaluating numerical techniques. By studying this controlled problem, the researchers examined whether an automatically discovered formula could improve the convergence behaviour of an established transport solution method.
The hidden challenge of mathematical convergence
In numerical analysis, convergence describes the process by which a sequence of approximations approaches a limiting value. A numerical method may generate increasingly accurate estimates as its calculation progresses, but the rate of improvement can vary considerably with the problem being solved.
Convergence acceleration techniques attempt to obtain a more accurate estimate from an existing sequence of approximations without requiring the same degree of additional computation. Rather than simply extending a calculation indefinitely, these techniques exploit mathematical relationships among previously calculated values to estimate the limiting solution more efficiently.
Traditional acceleration methods, however, often rely on assumptions about the behaviour of the sequence. Aitken’s delta-squared method is particularly effective for certain sequences that converge geometrically, while the Wynn-epsilon algorithm can perform well for alternating or oscillatory sequences. Their performance depends on how closely the actual convergence pattern matches the mathematical structure for which each method is suited.
Neutron transport solutions can exhibit different convergence characteristics depending on physical parameters, including the scattering ratio, the angular discretisation and the location at which the solution is evaluated. A method that performs well under one set of conditions may be less effective under another. This variability motivated the researchers to investigate whether a mathematical expression could be discovered directly from transport solution sequences instead of relying exclusively on conventional acceleration formulas.
Genetic programming applied to mathematics
Genetic programming (GP) is a computational technique inspired by biological evolution. Instead of starting with a fixed mathematical expression and adjusting its coefficients, the method searches for the structure of the expression itself. It can generate and evaluate alternative formulas, retaining promising candidates and modifying them over successive generations.
The process involves three principal operations: selection, crossover and mutation. Selection favours formulas that perform well according to a defined fitness function. Crossover combines parts of two existing formulas to produce new candidates, while mutation introduces random structural changes. An elitism mechanism preserves selected high-performing formulas so that their useful structures are not lost during subsequent generations.
The researchers used this evolutionary process to discover a formula tailored to the convergence characteristics of discrete ordinates \(S_N\) neutron transport solutions. Unlike conventional machine learning approaches that may produce predictions without an explicit mathematical expression, genetic programming through symbolic regression can identify a formula that can be examined and evaluated directly. This is particularly relevant to computational physics, where mathematical interpretability and numerical behaviour are important considerations.
How the researchers designed the experiment
The study investigated the steady-state, monoenergetic neutron transport equation in a homogeneous slab with isotropic scattering. The researchers generated 240 solution sequences using an analytical \(S_N\) method. The scattering ratio ranged from 0.001 to 0.999, while the quadrature order varied from \(N=4\) to \(N=52\), in increments of four.
In the discrete ordinates method, the angular domain is represented by a finite set of discrete directions and associated quadrature weights. Increasing quadrature order provides a sequence of increasingly refined angular approximations. As the order increases, the solutions approach the continuous angular limit for the specified transport problem. This sequence provides the numerical data on which convergence acceleration can operate.
The genetic programming system used consecutive sequence values and mathematical operations, including addition, subtraction, multiplication, division, squaring and second differences, to construct candidate expressions. The search was conducted in MATLAB 2025a, with a population of 40 candidate formulas per generation and a maximum of 200 generations. The researchers also seeded the initial population with a version of Aitken’s formula, allowing the evolutionary search to explore alternatives to an established method.
The dataset was divided into training and validation sets, with 168 sequences used for training and 72 held back for validation. The fitness function measured the fraction of training comparisons in which a candidate formula improved upon the unaccelerated sequence. The evolutionary search was configured to stop when a candidate achieved the target success rate of 75% or reached the generation limit. The reported run reached its target at generation 63.
A new formula outperforms established methods
The central finding was that the genetic programming approach discovered a rational mathematical expression incorporating cross-term products and second differences. These structural features distinguished it from the traditional acceleration techniques used for comparison. The search had the option of including a tunable constant but did not select one, indicating that the formula’s effectiveness derived entirely from the relationship between consecutive sequence values.
The evolved formula achieved a success rate of approximately 78% on the reported test comparisons. By contrast, Aitken’s delta-squared method achieved 39%, while the Wynn-epsilon algorithm achieved 32%. The results indicate that the automatically discovered expression was more successful at improving convergence than either conventional method for the problem set examined.
The validation results were also encouraging. The formula achieved approximately 78.1% success on the held-out validation data, with a difference of less than half a percentage point between training and validation performance. This small gap suggests that the result was not simply an improvement confined to the training sequences. However, the findings should be interpreted within the limits of the dataset and experimental configuration rather than as proof of universal generalisation.
Where the formula performed best, and where it failed
Each test sequence is a series of increasingly refined calculations. The researchers compared estimates from the formula with the unaccelerated solutions at five checkpoints in the series. At the last two checkpoints, the evolved formula won every comparison, and its typical error was about half that of the unaccelerated solutions. At the first checkpoint, it won nine times out of ten, and its typical error was about a third of the unaccelerated value. That advantage faded slightly in materials that scatter neutrons far more than they absorb them. At the middle checkpoint, it won almost every time, but only by a small margin. These differences show why a single overall success rate cannot fully describe how well a numerical method works.
The exception was the second checkpoint, where the formula never won, irrespective of the material. Because this failure did not depend on physics, the authors suggest it reflects a limitation built into the formula’s structure at that point in the sequence. Evolutionary discovery, in other words, does not automatically produce a method that is better in every circumstance. Understanding why the formula succeeds in some places and fails in others remains an open question.
Instead of giving the computer a predetermined equation and asking it to optimise a few parameters, we let the mathematical expressions themselves evolve.
—Japan K. Patel
Reference
Patel, J. K., Ganapol, B. D., Magliari, A., Schmidt, M. C., & Wareing, T. A. (2026). Evolutionary discovery of sequence acceleration methods for slab geometry neutron transport. Nuclear Science and Engineering. https://doi.org/10.1080/00295639.2026.2643119
