Article: Implicit neural representation and error control for solving mathematical partial differential equations Journal: International Journal of Information and Communication Technology (IJICT) 2026 Vol.27 No.55 pp.97 - 120 Abstract: 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<SUP align="right"><SMALL>-4</SMALL></SUP>, 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. Inderscience Publishers - linking academia, business and industry through research

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

Authors: Huaizhe Zhang

Addresses: Basic Teaching Department, Shanghai Zhongqiao Vocational and Technical University, Shanghai, 201514, China

Abstract: 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.

Keywords: partial differential equations; PDEs; adaptive optimisation; numerical simulation; implicit neural representation; INR.

DOI: 10.1504/IJICT.2026.153794

International Journal of Information and Communication Technology, 2026 Vol.27 No.55, pp.97 - 120

Received: 22 Jan 2026
Accepted: 24 Feb 2026
Published online: 26 May 2026 *

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

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