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Register a new userDeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators
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While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the potential application of neural networks in learning nonlinear operators from data. However, the theorem guarantees only a small approximation error for a sufficient large network, and does not consider the important optimization and generalization errors. To realize this theorem in practice, we propose deep operator networks (DeepONets) to learn operators accurately and efficiently from a relatively small dataset. A DeepONet consists of two sub-networks, one for encoding the input function at a fixed number of sensors $x_i, i=1,dots,m$ (branch net), and another for encoding the locations for the output functions (trunk net). We perform systematic simulations for identifying two types of operators, i.e., dynamic systems and partial differential equations, and demonstrate that DeepONet significantly reduces the generalization error compared to the fully-connected networks. We also derive theoretically the dependence of the approximation error in terms of the number of sensors (where the input function is defined) as well as the input function type, and we verify the theorem with computational results. More importantly, we observe high-order error convergence in our computational tests, namely polynomial rates (from half order to fourth order) and even exponential convergence with respect to the training dataset size.
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For a constant coefficient partial differential operator $P(D)$ with a single characteristic direction such as the time-dependent free Schrodinger operator as well as non-degenerate parabolic differential operators like the heat operator we characterize when open subsets $X_1subseteq X_2$ of $mathbb{R}^d$ form a $P$-Runge pair. The presented condition does not require any kind of regularity of the boundaries of $X_1$ nor $X_2$. As part of our result we prove that for a large class of non-elliptic operators $P(D)$ there are smooth solutions $u$ to the equation $P(D)u=0$ on $mathbb{R}^d$ with support contained in an arbitarily narrow slab bounded by two parallel characteristic hyperplanes for $P(D)$.
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The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutati
In this paper we develop the calculus of pseudo-differential operators corresponding to the quantizations of the form $$ Au(x)=int_{mathbb{R}^n}int_{mathbb{R}^n}e^{i(x-y)cdotxi}sigma(x+tau(y-x),xi)u(y)dydxi, $$ where $tau:mathbb{R}^ntomathbb{R}^n$ is a general function. In particular, for the linear choices $tau(x)=0$, $tau(x)=x$, and $tau(x)=frac{x}{2}$ this covers the well-known Kohn-Nirenberg, anti-Kohn-Nirenberg, and Weyl quantizations, respectively. Quantizations of such type appear naturally in the analysis on nilpotent Lie groups for polynomial functions $tau$ and here we investigate the corresponding calculus in the model case of $mathbb{R}^n$. We also give examples of nonlinear $tau$ appearing on the polarised and non-polarised Heisenberg groups, inspired by the recent joint work with Marius Mantoiu.
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