Unifying machine learning and interpolation theory via interpolating neural networks.

Computational science and engineering are shifting toward data-centric, optimization-based, and self-correcting solvers with artificial intelligence. This transition faces challenges such as low accuracy with sparse data, poor scalability, and high computational cost in complex system design. This work introduces Interpolating Neural Network (INN)-a network architecture blending interpolation theory and tensor decomposition. INN significantly reduces computational effort and memory requirements while maintaining high accuracy. Thus, it outperforms traditional partial differential equation (PDE) solvers, machine learning (ML) models, and physics-informed neural networks (PINNs). It also efficiently handles sparse data and enables dynamic updates of nonlinear activation. Demonstrated in metal additive manufacturing, INN rapidly constructs an accurate surrogate model of Laser Powder Bed Fusion (L-PBF) heat transfer simulation. It achieves sub-10-micrometer resolution for a 10 mm path in under 15 minutes on a single GPU, which is 5-8 orders of magnitude faster than competing ML models. This offers a new perspective for addressing challenges in computational science and engineering.