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I am experimenting with machine learning techniques to solve partial differential equations (PDEs). My goal is to use solutions from previous time steps to predict the solution at the next time step, essentially treating the problem as a multistep method.

However, when I applied this approach to the one-dimensional Kuramoto-Sivashinsky equation, I did not observe any improvement in the predictions compared to using only the current time step.

This led me to wonder if the issue lies with the specific equation I chose. Is there a class of PDEs that is particularly well-suited for multistep methods in the context of machine learning? I am looking for equations where using information from previous time steps can significantly enhance the accuracy of the predictions.

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    $\begingroup$ Can you give some more detail on what "no improvements in predications" entails? Any convergent time-stepping method can provide the same amount of error and the differences between methods are typically in how large a time step can be + how expensive a single step is while still staying under some error threshold $\endgroup$
    – whpowell96
    Commented May 2 at 14:41
  • $\begingroup$ @whpowell96 For the same temporal step size, I am measuring no improvements in error or correlation when I add more previous time steps. I haven't tried varying the step size. Keep in mind that this is a machine learning method, not a numerical method. $\endgroup$
    – user572780
    Commented May 2 at 14:46
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    $\begingroup$ A single-step method can be more accurate than multi-step methods for fixed step size. The differences come when you consider convergence order, truncation error coefficient, and various stability properties. $\endgroup$
    – whpowell96
    Commented May 2 at 14:49
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    $\begingroup$ You should fix the title of your question, which currently doesn't mention machine learning. Then you should add more details about your algorithm and what you're measuring. With the current amount of information, I don't think you will get a useful answer. $\endgroup$
    – David Ketcheson
    Commented May 3 at 6:18

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