Great question! Let’s walk through a clear, step-by-step approach to identify the unknown function $f(u,t)$ using your noise-free step response data, given the structure of your ODE.

First, let’s define each of your step inputs explicitly. For the k-th experiment, your input can be written as:
u(t) = U_k \cdot H(t)
where $U_k$ is the amplitude of the k-th step, and $H(t)$ is the unit step function (1 for $t \geq 0$, 0 otherwise). Plug this into your ODE:
$$\dot{x}_k(t) = \alpha f(U_k, t) + \beta U_k$$
Here, $x_k(t)$ is the measured step response corresponding to input amplitude $U_k$.

Since $\alpha$ is a known non-zero constant (otherwise the ODE reduces to a trivial linear system), we can rearrange the equation directly to isolate $f$:
$$f(U_k, t) = \frac{1}{\alpha} \left( \dot{x}_k(t) - \beta U_k \right)$$
This is the core relation—all we need now is the time derivative of your measured response data.

Your noise-free data makes this easier than dealing with noisy measurements. Here’s how to handle discrete or continuous data:

• Discrete sampled data: If you have measurements at time points $t_1, t_2, ..., t_N$ with interval $\Delta t$, use the central difference method for most points (best accuracy):
  $\dot{x}(t_i) = \frac{x(t_{i+1}) - x(t_{i-1})}{2\Delta t}$
  For the first and last points (where central difference isn’t possible), use forward/backward differences:
  $\dot{x}(t_1) = \frac{x(t_2) - x(t_1)}{\Delta t}$
$\dot{x}(t_N) = \frac{x(t_N) - x(t_{N-1})}{\Delta t}$
• Continuous data: If you have an analytic expression for $x(t)$ (or ultra-high-resolution samples), simply take the analytical derivative of $x(t)$.

With multiple step amplitudes, you’ll generate a set of $f(U_k, t)$ curves (one per $U_k$). Now you need to generalize these into a function that works for any $u$ and $t$:

1. Observe patterns: Look at how $f$ evolves with time (e.g., exponential decay, oscillation, polynomial trend) and how it scales with $u$ (e.g., linear, quadratic, power-law).
2. Choose a functional form: Based on your observations, pick a candidate form for $f(u,t)$. Common choices include:
   • Separable forms: $f(u,t) = g(t) \cdot h(u)$ (if time and input dependencies are independent)
   • Polynomial forms: $f(u,t) = \sum_{i=0}^{m} \sum_{j=0}^{n} a_{ij} t^i u^j$ (flexible for general nonlinearities)
3. Fit coefficients: Use a least-squares fitting method (since your data is noise-free, this should give an exact match if you choose the right form) to solve for the unknown coefficients (like $a_{ij}$ above).

Once you have a candidate $f(u,t)$, verify it by plugging it back into the original ODE:

1. Pick an arbitrary step amplitude $U$ (could be one of your existing experiments or a new one).
2. Simulate the ODE to generate $\dot{x}(t)$, then integrate to get $x(t)$.
3. Compare this simulated $x(t)$ to your measured response—since your data is noise-free, they should match exactly if your $f(u,t)$ is correct.

If $\alpha = 0$, your ODE simplifies to $\dot{x}(t) = \beta u(t)$, which means $f(u,t)$ doesn’t contribute to the dynamics (the problem becomes trivial). But based on your question framing, we can assume $\alpha \neq 0$.

内容的提问来源于stack exchange，提问作者Dr Krishnakumar Gopalakrishnan