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<p>We report the solution of a benchmark set of ordinary differential equations (ODEs) with genetic programming (GP) within a collocation framework using numerical tuning of the embedded tree constants. Alongside a conventional soft penalty formulation, we also report results from two GP variants that enforce the initial conditions on the ODEs as hard constraints: the first uses the so-called death penalty while the second employs a novel ranking method that orders infeasible individuals using Pareto dominance according to the degree to which they violate the constraints. We investigate the influence of the numbers of collocation points used to solve the problem, and conclude that a few ODEs require more than 10–20 points, otherwise the number of points is not critical. A statistical comparison of the different methods indicates that only a few ODEs display differences, an observation we attribute to the influence of parameter tuning. We obtain highly accurate solutions for all the benchmark ODEs, but identify a problem with certain of the ODEs producing trivial solutions, which we are able to mostly mitigate by introducing an additional constraint on the mean squared amplitude of the evolved solutions. Overall, we infer that the properties of the individual ODEs can impact the solution process.</p>

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Solving ordinary differential equations with genetic programming with hard initial/boundary value constraints

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年3月27日

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  • Peter Rockett

摘要

We report the solution of a benchmark set of ordinary differential equations (ODEs) with genetic programming (GP) within a collocation framework using numerical tuning of the embedded tree constants. Alongside a conventional soft penalty formulation, we also report results from two GP variants that enforce the initial conditions on the ODEs as hard constraints: the first uses the so-called death penalty while the second employs a novel ranking method that orders infeasible individuals using Pareto dominance according to the degree to which they violate the constraints. We investigate the influence of the numbers of collocation points used to solve the problem, and conclude that a few ODEs require more than 10–20 points, otherwise the number of points is not critical. A statistical comparison of the different methods indicates that only a few ODEs display differences, an observation we attribute to the influence of parameter tuning. We obtain highly accurate solutions for all the benchmark ODEs, but identify a problem with certain of the ODEs producing trivial solutions, which we are able to mostly mitigate by introducing an additional constraint on the mean squared amplitude of the evolved solutions. Overall, we infer that the properties of the individual ODEs can impact the solution process.