Transformers are deep architectures that define ``in-context mappings'' which enable predicting new tokens based on a given set of tokens (such as a prompt in NLP applications or a set of patches for a vision transformer). In this work, we study in particular the ability of these architectures to handle an arbitrarily large number of context tokens. To mathematically, uniformly address their expressivity, we consider the case that the mappings are conditioned on a context represented by a probability distribution of tokens which becomes discrete for a finite number of these. The relevant notion of smoothness then corresponds to continuity in terms of the Wasserstein distance between these contexts. We demonstrate that deep transformers are universal and can approximate continuous in-context mappings to arbitrary precision, uniformly over compact token domains. This result implies, as a special case, that transformers are universal approximators for continuous permutation-invariant mappings over a fixed number of tokens. It also establishes the universal approximation capability of transformers for certain in-context learning tasks, demonstrating in particular their ability to perform regression within context. A key aspect of our results, compared to existing findings, is that for a fixed precision, a single transformer can operate on an arbitrary (even infinite) number of tokens. Additionally, it operates with a fixed embedding dimension of tokens (this dimension does not increase with precision) and a fixed number of heads (proportional to the dimension). The use of MLPs between multi-head attention layers is also explicitly controlled. We consider both unmasked attentions (as used for the vision transformer) and masked causal attentions (as used for NLP and time series applications). We tackle the causal setting leveraging a space-time lifting to analyze causal attention as a mapping over probability distributions of tokens.

Neural operators serve as universal approximators for general continuous operators. In this paper, we derive the approximation rate of solution operators for the nonlinear parabolic partial differential equations (PDEs), contributing to the quantitative approximation theorem for solution operators of nonlinear PDEs. Our results show that neural operators can efficiently approximate these solution operators without the exponential growth in model complexity, thus strengthening the theoretical foundation of neural operators. A key insight in our proof is to transfer PDEs into the corresponding integral equations via Duahamel's principle, and to leverage the similarity between neural operators and Picard’s iteration—a classical algorithm for solving PDEs. This approach is potentially generalizable beyond parabolic PDEs to a class of PDEs which can be solved by Picard's iteration.

$1$-Lipschitz neural networks are fundamental for generative modelling, inverse problems, and robust classifiers. In this paper, we focus on $1$-Lipschitz residual networks (ResNets) based on explicit Euler steps of negative gradient flows and study their approximation capabilities. Leveraging the Restricted Stone–Weierstrass Theorem, we first show that these $1$-Lipschitz ResNets are dense in the set of scalar $1$-Lipschitz functions on any compact domain when width and depth are allowed to grow. We also show that these networks can exactly represent scalar piecewise affine $1$-Lipschitz functions. We then prove a stronger statement: by inserting norm-constrained linear maps between the residual blocks, the same density holds when the hidden width is fixed. Because every layer obeys simple norm constraints, the resulting models can be trained with off-the-shelf optimisers. This paper provides the first universal approximation guarantees for $1$-Lipschitz ResNets, laying a rigorous foundation for their practical use.

In this work we consider the problem of discretization of neural operators in a general setting. Using category theory, we give a no-go theorem that shows that diffeomorphisms between Hilbert spaces may not admit any continuous approximations by diffeomorphisms on finite spaces, even if the discretization is non-linear. This shows how infinite-dimensional Hilbert spaces and finite-dimensional vector spaces fundamentally differ. A key take-away is that to obtain discretization invariance, considerable effort is needed to ensure that finite-dimensional approximations of neural operator converge not only as sequences of functions, but that their representations converge in a suitable sense as well. With this perspective, we give several positive results. We first show that strongly monotone diffeomorphism operators always admit finite-dimensional strongly monotone diffeomorphisms. Next we show that bilipschitz neural operators may always be written via the repeated alternating composition of strongly monotone neural operators and invertible linear maps. We also show that such operators may be inverted locally via iteration provided that such inverse exists. Finally, we conclude by showing how our framework may be used `out of the box' to prove quantitative approximation results for discretization of neural operators.

Recently there has been great interest in operator learning, where networks learn operators between function spaces from an essentially infinite-dimensional perspective. In this work we present results for when the operators learned by these networks are injective and surjective. As a warmup, we combine prior work in both the finite-dimensional ReLU and operator learning setting by giving sharp conditions under which ReLU layers with linear neural operators are injective. We then consider the case when the activation function is pointwise bijective and obtain sufficient conditions for the layer to be injective. We remark that this question, while trivial in the finite-rank setting, is subtler in the infinite-rank setting and is proven using tools from Fredholm theory. Next, we prove that our supplied injective neural operators are universal approximators and that their implementation, with finite-rank neural networks, are still injective. This ensures that injectivity is not 'lost' in the transcription from analytical operators to their finite-rank implementation with networks. Finally, we conclude with an increase in abstraction and consider general conditions when subnetworks, which may have many layers, are injective and surjective and provide an exact inversion from a 'linearization.’ This section uses general arguments from Fredholm theory and Leray-Schauder degree theory for non-linear integral equations to analyze the mapping properties of neural operators in function spaces. These results apply to subnetworks formed from the layers considered in this work, under natural conditions. We believe that our work has applications in Bayesian uncertainty quantification where injectivity enables likelihood estimation and in inverse problems where surjectivity and injectivity corresponds to existence and uniqueness of the solutions, respectively.

Recurrent neural networks (RNNs) are a class of neural networks used in sequential tasks. However, in general, RNNs have a large number of parameters and involve enormous computational costs by repeating the recurrent structures in many time steps. As a method to overcome this difficulty, RNN pruning has attracted increasing attention in recent years, and it brings us benefits in terms of the reduction of computational cost as the time step progresses. However, most existing methods of RNN pruning are heuristic. The purpose of this paper is to study the theoretical scheme for RNN pruning method. We propose an appropriate pruning algorithm for RNNs inspired by "spectral pruning", and provide the generalization error bounds for compressed RNNs. We also provide numerical experiments to demonstrate our theoretical results and show the effectiveness of our pruning method compared with the existing methods.

One span of an aqueduct bridge suddenly collapsed in Wakayama City in western Japan on October 3, 2021. This study investigates the use of remote sensing data for the assessment of bridge situations in the normal time and in accidents/disasters. A field survey was conducted by the authors with the aid of a small UA V. Google Street View photos taken before the accident were also used. Based on these data, it is estimated that more than 5 hangers out of 18 might have failed for the collapsed span when the bridge collapse occurred. The failure of 4 hangers was also confirmed in the adjacent surviving span from the UAV images. The pre- and post-event high-resolution TerraSAR- X intensity images were also introduced to extract the collapsed span from the SAR data.