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by ChrisRackauckas
2123 days ago
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For reference, the DiffEqFlux library has a bunch of classical basis layers [1] and ways to tensor product them [2] for this reason. The real answer as to when to use a neural network is quite complicated [3], but in summary the results all point to the fact that for approximating an R^n -> R^m function, one only needs polynomially many parameters in order to do it well (as proven in a few cases like in that linked paper for "any case where Monte Carlo algorithms are not exponential in dimension"). Tensor products of classical basis functions have to cover every combination of terms (sin(ix)*sin(jy)) so they naturally grow like p^n if you have p parameters in each dimension, so this exponential parameter growth is the curse of dimensionality and this polynomial growth is the formal way of describing how neural networks overcome the curse of dimensionality. So what is useful can depend on a number of factors (another property is the isotropy of the function you're trying to approximate), but this asymptotic property is what makes neural networks a good tool in the high dimensional world where they are commonly used. That makes them quite good as well for things like feedback controllers of larger ODE systems. But yes, in smaller dimensional cases Fourier basis and such are good choices. [1] https://diffeqflux.sciml.ai/dev/layers/BasisLayers/
[2] https://diffeqflux.sciml.ai/dev/layers/TensorLayer/
[3] https://arxiv.org/abs/1908.10828
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