神经算子是否总能被连续离散化？
Can neural operators always be continuously discretized?
AI总结：
针对神经算子能否连续离散化的问题，论文证明一般无限维微分同胚存在不可能性，并提出强单调与逐层强单调神经算子以保证离散化不变性及表示收敛，还给出双 Lipschitz 神经算子的分解、有限秩逼近与定量结果。
摘要：
我们在一个包含跳跃连接的通用框架中，研究 Hilbert 空间之间神经算子的离散化问题。我们通过无限维微分同胚的视角，重点关注双射神经算子。借助范畴论进行表述，我们给出了一个不可能性定理，表明 Hilbert 空间或 Hilbert 流形之间的微分同胚，可能无法由有限维空间上的微分同胚给出任何连续逼近，即使这些逼近是非线性的也是如此。自然的出路是引入强单调微分同胚和逐层强单调神经算子，它们能够由有限维空间上的强单调微分同胚连续逼近。对于这些对象，我们可以保证离散化不变性，同时确保有限维逼近不仅作为函数序列收敛，而且其表示也在适当意义下收敛。最后，我们证明双 Lipschitz 神经算子总可以写成强单调神经算子的交替复合，再加上一个简单等距变换的形式。由此，我们为一类广义神经算子的离散化建立了严格平台。我们还表明，此类神经算子可以通过有限秩残差神经算子的复合来逼近，其中每个块都是强单调的，并且可以通过迭代进行局部求逆。最后，我们为一般双 Lipschitz 神经算子的离散化提供了一个定量逼近结果。
英文摘要：
We consider the problem of discretization of neural operators between Hilbert spaces in a general framework including skip connections. We focus on bijective neural operators through the lens of diffeomorphisms in infinite dimensions. Framed using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces or Hilbert manifolds may not admit any continuous approximations by diffeomorphisms on finite-dimensional spaces, even if the approximations are nonlinear. The natural way out is the introduction of strongly monotone diffeomorphisms and layerwise strongly monotone neural operators which have continuous approximations by strongly monotone diffeomorphisms on finite-dimensional spaces. For these, one can guarantee discretization invariance, while ensuring that finite-dimensional approximations converge not only as sequences of functions, but that their representations converge in a suitable sense as well. Finally, we show that bilipschitz neural operators may always be written in the form of an alternating composition of strongly monotone neural operators, plus a simple isometry. Thus we realize a rigorous platform for discretization of a generalization of a neural operator. We also show that neural operators of this type may be approximated through the composition of finite-rank residual neural operators, where each block is strongly monotone, and may be inverted locally via iteration. We conclude by providing a quantitative approximation result for the discretization of general bilipschitz neural operators.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
