# Fitting nonlinear differential equations to correspond a predefined solution

When modeling temporal dynamics of a biological process I stumbled upon a set of differential equations having the following matrix form: $$\frac{d\mathbf{a}(t)}{dt}=\Gamma(t)\mathbf{a}(t)+C\mathbf{a}(t),$$ where $\mathbf{a}(t)=(a_1(t),\ldots,a_N(t))^T$ is the column of the functions to be found, $\Gamma(t)=\text{diag}(\gamma_1(t),\ldots,\gamma_N(t))$ is the diagonal matrix with given function entries, $C$ is the constant matrix describing nearest-neibor interaction: $$C= \left(\begin{matrix} 0 & c & \ldots& 0 & 0\\ c & 0 & \ldots & 0 & 0\\ \vdots & \vdots& \ddots& \vdots & \vdots\\ 0 & 0 & \ldots & 0 & c\\ 0 & 0 & \ldots & c & 0 \end{matrix}\right).$$

Provided that the matrices $\Gamma(t)$ and $C$ are defined and the initial conditions are specified: $$\mathbf{a}(t=0)=\mathbf{a}_0,$$ the equation set can be solved by a suitable method, at least numerically.

However, my actula task is inverse. Namely, instead of specifiying matrix $\Gamma(t)$, I have a solution of the equations set at a time $t_1>t_0$, which is written through the initial values by the matrix $$T= \left(\begin{matrix} T_{11} & T_{12} & \ldots & T_{1N}\\ T_{21} & T_{22} & \ldots & T_{2N}\\ \vdots & \vdots & \ddots & \vdots\\ T_{N1} & T_{N2} & \ldots & T_{NN} \end{matrix}\right),$$ so that $$\mathbf{a}(t_1)=T\mathbf{a}_0.$$
In addition to the diagonal form, there is a kind of conservation law the elements of the functional matrix $\Gamma(t)$ should obey: $$\sum_{i=1}^Na_i(t)\gamma_i(t)=E,$$ with $a_i(t)$ being known functions and $E>0$ is a free parameter, which can be chosen independently for each solution $T$.

I wonder, if I can choose proper matrix $\Gamma(t)$ with smooth constituent functions to correspond each solution $T$ or not? Where can I find a mathematical proof if any?

I understand, that the equations are generally solved numerically, so done the inverse problem. Therefore, I am kindly asking you to hint a numerical procedure to handle the problem.

• Actually, last year I've seen many articles about inferring laws of motion from observable data. However, that was out of my scope of interest, but I think that digging in Google Scholar might give you few useful papers. However, I think that some prior information about unknown functions is needed for efficient solving of problem. – Evgeny Dec 22 '15 at 19:29
• Could you give me some keywords to search for? I am not working in the field of mathematics and, thus, I don't know with certainty the specific area the task relates to. – MikhailShmulevich Dec 22 '15 at 23:11
• I think something like "system identification learning in dynamical systems" would be nice for a start. I also recommend to try "dynamical systems machine learning". – Evgeny Dec 23 '15 at 0:36