Implicit neural representation and error control for solving mathematical partial differential equations

This research tackles the persistent challenge of uncontrolled approximation errors and unreliable convergence in neural network-based methods for solving partial differential equations.We introduce a novel error-controlled implicit neural representation framework, which incorporates a trainable error indicator network and an adaptive weighting scheme to dynamically steer the optimisation process.Our approach utilises a dual-encoding architecture to represent physical fields with high fidelity and a cooperative training mechanism that iteratively estimates and reduces local errors.Experimental validation on a national aeronautics and space administration turbulent flat-plate boundary layer benchmark demonstrates that error-controlled implicit neural representation achieves a relative L2 error of 8.73 × 10 -4 , outperforming the best existing baseline by 42.6% and improving boundary-layer accuracy by 52.0%.Moreover, the proposed method reduces training time by 34.7%-55.2%while maintaining physically consistent solutions, confirming its efficacy and efficiency in error-aware numerical simulation.