# Functional approximation with derivatives

I am trying to solve a functional approximation problem.

Consider a set of measurements of a d-dimensional state $$\mathrm x \in \mathbb{R}^d$$, together with velocities $$\dot{\mathrm x}$$ and accelerations $$\ddot{\mathrm x}$$.

From some theoretical derivations I have obtained the following equality from which I want to estimate/approximate the function $$f(\mathrm x):\mathbb{R}^d \rightarrow \mathbb{R}^d$$

$$\ddot{\mathrm x} = -J^{\top}_f(\mathrm x) \left[ \dot{\mathrm x} - f(\mathrm x) \right] + 2 [CC^{\top}]^{-1} \nabla\left( \nabla \cdot f(\mathrm x) \right), \;\;\;\;\;\;\;\;\;\;\;(1)$$

where $$J_f(\mathrm x)$$ denotes the Jacobian of $$f(x)$$, and $$C$$ is a $$d \times d$$ matrix.

In principle for every measurement $$(\mathrm x, \dot{\mathrm x}, \ddot{\mathrm x})$$ I can formulate one of these equations, represent $$f(\mathrm x)$$ with basis functions, and solve for the coefficients. I.e. I would assume risk $$\mathcal{L} = \frac{1}{2} \| ( \ddot{x} - \hat{\ddot{x}})^{\top} ( \ddot{x} - \hat{\ddot{x}}) \|,$$ where with $$\hat{\ddot{x}}$$ I denote the rhs of $$(1)$$, represent the function f in terms of some basis $$f(\mathrm x) = \sum^K_{k=1} a_k \phi_k(\mathrm x)$$, and solve for the coefficients by setting $$\nabla_{\mathrm a} \mathcal{L} = 0$$.

Right?

Hence this expression looks far too complicated since I will have up to 4th order products of coefficients $$a_k$$.

Is there any simpler way to approximate $$f(\mathrm x)$$?? I.e. neural networks or some other method that I might not be aware of?