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?
