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For surrogate models which predict derivatives based on the current state: how do you avoid the accumulation of state errors due to modeling error in each state update?

It seems to me that if you used a surrogate model (predicting derivatives) for e.g. turbulent simulations. Then by definition of chaotic dynamics/fixed points: any error would compound exponentially. Which actually means the problem would be two-fold: the error in the model's state input would exacerbate modeling error for future derivative predictions & any non-trivial amount of error would compound on it's own anyways (even with perfect state updates) due to chaos.

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While ROMs and other surrogate models can have exponentially growing error in the worst case (the few rigorous error bounds are like this), a surrogate that is stable will only exhibit these exponentially growing errors for short times. Such a model would hopefully evolve on another attractor, close to the original in state space, so the distance between states is bounded from above. The goal for these types of problems is to construct or constrain the surrogate in a way that it can reproduce or approximate the chaotic parts of the system while the state space residual remains small.

That being said, the cruxes are

  1. Whether the surrogate model is stable. Even very simple surrogate modeling methods such as proper orthogonal decomposition can become unstable for nonlinear systems under very small changes in model parameters. There is lots of current research of ROM stabilization, which tweaks the ROM basis to ensure that parts of the solution cannot blow up. However, many factors can lead to ROM instability and there is no general practice for fixing it.

  2. Whether the surrogate is accurate. Depending on the sophistication of the surrogate model and the original system, there is often very little in the way of rigorous a priori or a posteriori error bounds or indicators, so a lot of surrogate development is in encoding heuristics and desireable properties into the surrogate structure in hopes that it reproduces key details of the original dynamics. I have seen some rigorous error bounds for POD ROMs on some general systems, but these bounds are often incredibly pessimistic and not informative.

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  • $\begingroup$ Thanks for the answer. But chaotic dynamics (e.g. turbulence) are mathematically defined by sensitivity to IC's, so even a small amount of error introduced only at one point in time should cause exponential divergence between surrogate & true trajectory. IMO this implies any error is unacceptable. So what am I missing here? $\endgroup$
    – profPlum
    Commented Jan 10 at 19:29
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    $\begingroup$ Exponential divergence of nearby trajectories is bounded above by the diameter of the attractor. It is still possible to get valuable information about the chaotic attractor even if you know the trajectory you are simulating diverged from the "true" trajectory long ago, as long as you are confident the attractor you are simulating is close to the truth. Even industrial-scale CFD applications don't trust the time accuracy of turbulent simulations, but they still trust the statistics from the simualted attractor. $\endgroup$
    – whpowell96
    Commented Jan 10 at 19:33
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    $\begingroup$ Thanks, this comment was very insightful. $\endgroup$
    – profPlum
    Commented Jan 10 at 19:41

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