Adaptive Error-Guided Hybrid Numerical–AI Framework for Nonlinear Multiscale PDE Systems

Abstract

Nonlinear multistate systems play a crucial role in a broad variety of science fields, such as fluid dynamics, heat transfer, reaction diffusion and others, wherein intricate interactions that span multiple spatial and temporal scales are particularly challenging to compute. Finite element and finite difference techniques are traditional examples of numerical techniques that provide high accuracy, but are computationally infeasible when the fine scale dynamics are needed. In contrast, new data-driven methods involving machine learning are computationally efficient, but may not be physically consistent or robust, especially in highly nonlinear regimes. In a bid to overcome these shortcomings, this paper presents an adaptive hybrid numerical-AI method of solving nonlinear multistate partial differential equations. The suggested method combines a physics-based finite element coarse-scale approximation solver with a neural operator-based model to model fine-scale corrections. One such innovation is an error-driven adaptive refinement mechanism, which involves dynamically adapting the computational resources through the evaluation of the residual of the governing equations and selectingive refinement of areas where error is large. This close-knit between the numerical and learning elements guarantees physical faithfulness and computational efficiency. The framework is tested on representative nonlinear multistate systems, such as Burgers equation, the NavierStokes flow and reaction diffusion models. Findings indicate high accuracy gains, more L2 error and less residual error are reduced as compared to single numerical and machine learning methods, with a high speed up and improved data efficiency as well. The suggested approach is continually stable and generalized under different boundary conditions and scale differences. On the whole, this work defines a scalable, physically consistent computational framework that fills in the gaps between traditional numerical simulation and data-driven modeling, and provides a pathway to next-generation scientific computing of complex multistate systems.