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28885 results in Differential and integral equations, dynamical systems and control theory

Page 45 of 1445

  • From NeurODEs to AutoencODEs: A mean-field control framework for width-varying neural networks

  • Part of
  • Cristina Cipriani, Massimo Fornasier, Alessandro Scagliotti
  • Journal:
    European Journal of Applied Mathematics / Volume 36 / Issue 2 / April 2025
    Published online by Cambridge University Press:
    08 February 2024, pp. 188-230
    • Article
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  • The connection between Residual Neural Networks (ResNets) and continuous-time control systems (known as NeurODEs) has led to a mathematical analysis of neural networks, which has provided interesting results of both theoretical and practical significance. However, by construction, NeurODEs have been limited to describing constant-width layers, making them unsuitable for modelling deep learning architectures with layers of variable width. In this paper, we propose a continuous-time Autoencoder, which we call AutoencODE, based on a modification of the controlled field that drives the dynamics. This adaptation enables the extension of the mean-field control framework originally devised for conventional NeurODEs. In this setting, we tackle the case of low Tikhonov regularisation, resulting in potentially non-convex cost landscapes. While the global results obtained for high Tikhonov regularisation may not hold globally, we show that many of them can be recovered in regions where the loss function is locally convex. Inspired by our theoretical findings, we develop a training method tailored to this specific type of Autoencoders with residual connections, and we validate our approach through numerical experiments conducted on various examples.

  • Regularity and linear response formula of the SRB measures for solenoidal attractors

  • Part of
  • CARLOS BOCKER, RICARDO BORTOLOTTI, ARMANDO CASTRO
  • Journal:
    Ergodic Theory and Dynamical Systems / Volume 44 / Issue 10 / October 2024
    Published online by Cambridge University Press:
    06 February 2024, pp. 2782-2831
    Print publication:
    October 2024

  • We show that a class of higher-dimensional hyperbolic endomorphisms admit absolutely continuous invariant probabilities whose densities are regular and vary differentiably with respect to the dynamical system. The maps we consider are skew-products given by $T(x,y) = (E (x), C(x,y))$, where E is an expanding map of $\mathbb {T}^u$ and C is a contracting map on each fiber. If $\inf |\!\det DT| \inf \| (D_yC)^{-1}\| ^{-2s}>1$ for some ${s<r-(({u+d})/{2}+1)}$, $r \geq 2$, and T satisfies a transversality condition between overlaps of iterates of T (a condition which we prove to be $C^r$-generic under mild assumptions), then the SRB measure $\mu _T$ of T is absolutely continuous and its density $h_T$ belongs to the Sobolev space $H^s({\mathbb {T}}^u\times {\mathbb {R}}^d)$. When $s> {u}/{2}$, it is also valid that the density $h_T$ is differentiable with respect to T. Similar results are proved for thermodynamical quantities for potentials close to the geometric potential.

Page 45 of 1445