This work extends kinetic-based regularization (KBR), a localized one-parameter kernel regression method, to spatial-derivative estimation. An improved correction makes the local quadratic fit second-order accurate away from training points. An explicit scheme extracts derivatives from closed-form fitted coefficients; an implicit scheme perturbs a query point and solves a small local system. On one-dimensional Camel and Rastrigin functions, explicit extraction is more stable for clean unseen data, while implicit extraction is more robust to noise. Clean-data tests outperform a differential-neural-network baseline and compare favorably with nonuniform finite differences. A preliminary two-dimensional example recovers gradients and Hessians. KBR is also inserted into conservative solvers through interface-flux prediction for inviscid Burgers and Sod shock-tube problems. MacCormack- and Roe-based variants remain stable and preserve conservative structure, although the Burgers solution exhibits Gibbs oscillations and irregular high-dimensional point clouds remain future work.
- Corrected KBR's local quadratic reconstruction to attain the intended accuracy at unseen query points.
- Derived explicit and implicit procedures for extracting gradients and second derivatives.
- Characterized complementary behavior: stronger clean-data stability for the explicit scheme and greater noise robustness for the implicit scheme.
- Demonstrated preliminary two-dimensional gradient and Hessian recovery.
- Integrated KBR interface-flux predictions into conservative MacCormack and Roe shock-capturing solvers.