My question is, what are the limitations of approximating a high dimensional nonlinear system of ODEs or PDEs with a lower dimensional linear system of ODEs. I understand that this is a big question, so any answer will have limited generality.
I was looking at an interesting video by Steve Brunton on Dynamic Mode Decomposition, where he looks at taking a high dimensional nonlinear fluid dynamics PDE model and tries to represent it as a lower dimensional system. Embedded within the first step of the DMD procedure is taking the SVD of the data matrix from the PDE and obtaining the first 5 or so singular vectors for the the system. So the DMD procedure represents the original nonlinear system with a linear system derived from the singular vectors.
Now this is not my area of expertise, but I was not sure about the limitations of representing a nonlinear system with a linear system. Approximation theory will often do something like this, where it represents some complicated system as the linear sum of some simple basis functions or kernel functions, such as a Fourier series, radial basis functions, or gaussian kernels, etc.
My question really stems around fixed point behavior in the nonlinear system. As we get to higher dimensional systems of ODEs and PDEs, do we really worry about fixed points any more? I say that for two reasons.
First, in the linear approximation to the nonlinear system, it is easy to obtain fixed points for the system by solving the system $Ax = b$, where $b$ is the zero vector. But I am not clear if the null space of the linear system matches any meaningful fixed points in the nonlinear system?
Second, as we deal with higher dimensional systems, such as 10,000 or 100,000 dimensional models from chemistry, or financial markets, then can we even find the fixed points? Let's assume for the sake of argument, that fixed points exist in the system. This is a very strong assumption, but let's assume it any way. To find the fixed points of this system, I would have to numerically search for roots of the first derivative of the system. So even if some fixed points exist, do numerical root finding methods exist that will work in even a moderate number of dimensions? I seem to remember that there are no global guarantees for numerical root finding even in a small number of dimensions for a nonlinear system. Hence in more dimensions the curse of dimensionality would further impair numerical root finding methods, right?
Hence I am not clear on how well a linear system of ODEs can approximate a higher dimensional nonlinear system of ODEs or PDEs. I was hoping someone could validate the logic above--or invalidate it if I am making some mistake.