Local Basis Neural Fields: Scalable Scientific Signal Fitting with Explicit Order of Accuracy for Values and Derivatives

Neural field representations are increasingly used to represent, visualize, and analyze large-scale scientific signals, such as astronomical simulations and turbulent flows. These applications demand scalable representations that are accurate not only in reconstructed values, but also under differentiation. However, purely MLP-based networks scale poorly to large signals, while hybrid methods that pair MLPs with grid-based features (e.g., hash grids) introduce significant artifacts under differentiation. To bridge this gap, we introduce Local Basis Neural Fields (LBNFs), a framework that represents a signal using a set of overlapping local polynomial functions placed adaptively across the domain, paired with a lightweight MLP that decodes these local features into signal values. The polynomial degree and the placement density together give LBNFs explicit, independent control over modeling complexity and its allocation around complex signal structures. Because LBNFs build on classical partition-of-unity (PoU) approximation theory, they inherit its convergence guarantees: for smooth targets, this yields $O(h^{p+1})$ convergence in value and $O(h^{p+1-\nu })$ convergence for derivatives of order $\nu \le p$, where $h$ is the spacing between polynomials and $p$ their degree. Empirically, LBNFs realize these $hp$-approximation rates on analytic benchmarks while achieving significantly lower absolute error than classical approximants. On large-scale scientific data, LBNFs scale comparably to state-of-the-art neural field methods (e.g., Instant-NGP and NeuRBF), matching or improving their reconstruction error while reducing first- and second-order gradient errors by an order of magnitude.