An Adaptive Multi-Interval Numerical Method With Neural-Guided Mesh Refinement for Parametric ADR Equations

We propose a parametric deep-learning-assisted numerical framework for the efficient and robust solution of nonlinear one-dimensional advection-diffusion-reaction (ADR) equations, with particular emphasis on strongly advection-dominated regimes characterized by thin boundary layers. The framework seamlessly integrates a parameter-geometry adaptive deep neural network (PGA-DNN) with a high-fidelity numerical solver. The PGA-DNN is formulated as a feedforward architecture with multiple hidden layers and nonlinear activation functions, and is trained in a cyclic, data-driven manner using solution data generated by the solver. This design enables a flexible and adaptive representation across varying parameter regimes and mesh configurations. To overcome the incompatibility of classical Chebyshev spectral collocation (ChSC) time discretization with time-dependent, nonuniform spatial meshes, we construct an implicit-explicit local differential transform method combined with a multi-interval Chebyshev collocation scheme (IELDTM-MIChSCM). The solver is further extended to incorporate spatial adaptivity, resulting in a fully adaptive AIELDTM-MIChSCM formulation. By explicitly treating the diffusion coefficient as an input parameter, the proposed framework efficiently learns entire parametric families of solutions within a single training process, thereby eliminating the need for retraining across different diffusion regimes. During the adaptive stage, physics-informed residuals guide the construction of time-dependent, nonuniform spatial meshes on each time subinterval. These meshes are directly embedded within the AIELDTM-MIChSCM solver, while the resulting high-fidelity numerical solutions are utilized iteratively to enrich the neural network training data set. Extensive numerical experiments on representative nonlinear ADR benchmark problems demonstrate that the proposed approach consistently achieves superior accuracy and computational efficiency compared with several established numerical methods, particularly in regimes with small diffusion parameters. Collectively, these results highlight that the integration of neural network-driven parametric and geometric adaptivity with a Taylor-series-based numerical solver constitutes a powerful, efficient, and robust paradigm for the approximation of parametric partial differential equations exhibiting multiscale behaviour.