A Neural Residual Correction of the Explicit Euler Method via Learned Truncation Error Operators: A Lotka–Volterra Case Study

The explicit Euler method is widely used for the numerical integration of ordinary differential equations due to its simplicity and low computational cost; however, its accuracy and long-term qualitative behavior deteriorate for nonlinear oscillatory systems when moderate or large time steps are employed. This work proposes a neural residual correction framework for the Lotka–Volterra system, where a neural network learns a data-driven approximation of the local truncation error of the Euler scheme relative to a sixth-order Taylor reference solution (Taylor6). The learned correction is incorporated into the Euler update, yielding a hybrid integrator that preserves the simplicity of the base method while improving accuracy and long-term boundedness. Separate neural networks are trained for fixed time-step sizes, with emphasis on an intermediate regime ($\Delta t\approx 0.8$–$1.1$) where the standard Euler method exhibits pronounced qualitative distortions. Numerical experiments show that the corrected method reduces phase and amplitude errors, preserves the qualitative structure of phase portraits, and generalizes to previously unseen parameter configurations. Comparisons with the classical fourth-order Runge–Kutta method further illustrate the robustness of the proposed approach in this regime. Unlike previous neural correction approaches focused on small time steps, high-order solvers, or global solution operators, the proposed framework specifically targets the intermediate regime where Euler’s truncation error becomes qualitatively dominant while remaining sufficiently structured to be learned through residual correction. Throughout this work, the term “stability” is used exclusively in a qualitative sense, referring to long-term boundedness and phase-portrait preservation.