I am solving an implicit function f(x, y(x)) = 0 with x, y \in \mathbb{R}^n (typical n is 50–200), using iterative methods (trust region), at many x’s. This question is about finding a good initial value y_0(x) for each x, good initial values speed up the solution. Suppose I have solved the prob…

[image] Tamas_Papp: Otherwise, all the models I am familiar with use just values, not derivatives. It feels like a waste not to use them since they are already computed. Another choice is the Gradient-Enhanced Neural Network (GENN), GEKPLS is generally the preferred method since pure NNs are …

[image] Tamas_Papp: A naive way I thought of is obtaining first-order approximations from all points, y^{(i)}(x) = y_i + \partial y/\partial x \cdot (x - x_i) and averaging them out somehow. But this looks very ad-hoc, and putting discipline on this would be nice, surely someone has invented a…

[image] Tamas_Papp: with x, y \in \mathbb{R}^n (typical n is 50–200) […] Importantly, typically k < n: I have fewer points than dimensions. Dimensions that high seem to rule out any kind of tensor-product method like Hermite polynomials or splines, and make interpolation difficult in general.…

[image] stevengj: Another option is radial basis functions (RBFs), and there are “Hermite RBF” variants that exploit derivative information, e.g. Vaillant (2013) or Fashamiha and Salac (2025) . GEKPLS is Gradient-Enhanced Kriging, i.e. a Hermite RBF.

Thanks for all the answers! After experimenting a bit with my problem I think that my main problem will be with extrapolation, not interpolation. That is, often the point at which I want a surrogate will be outside (the convex hull of) the existing points. What seems to work decently in practice (…

[image] ChrisRackauckas: GEKPLS is Gradient-Enhanced Kriging, i.e. a Hermite RBF. Good point, although RBFs (unlike Kriging) are traditionally augmented with some low-order polynomials, and often use a wider variety of basis functions than Gaussians.

Surrogate with derivatives

Specific Domains Modelling & Simulations

ChrisRackauckas May 26, 2026, 8:41am 5

GEKPLS is Gradient-Enhanced Kriging, i.e. a Hermite RBF.

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Surrogate with derivatives

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