Ali Syed

Shao-Ting Chiu, Aditya Nambiar, Ali Syed et al.

An important application of neural networks to scientific computing has been the learning of non-linear operators. In this framework, a neural network is trained to fit a non-linear map between two infinite dimensional spaces, for example, the solution operator of ordinary and partial differential equations. Recently, inspired by the discovery of in-context learning for large language models, an even more ambitious paradigm has been explored, called multi-operator learning. In this approach, a neural network is trained to learn many different operators at the same time. In order to evaluate one of the learned operators, the network is passed example inputs and outputs to disambiguate the desired operator. In this work, we provide a precise mathematical formulation of the multi-operator learning problem. In addition, we modify a simple efficient architecture, called DeepOSets, for multi-operator learning and prove its universality for multi-operator learning. Finally, we provide a comprehensive set of experiments that demonstrate the ability of DeepOSets to learn multiple operators corresponding to different initial-value and boundary-value differential equations and use in-context examples to predict accurately the solutions corresponding to queries and differential equations not seen during training. The main advantage of DeepOSets is its architectural simplicity, which allows the derivation of theoretical guarantees and training times that are in the order of minutes, in contrast to similar transformer-based alternatives that are empirically justified and require hours of training.

Mahmut Yurt, Kanghyun Ryu, Zhitao Li et al.

Conventional cardiac cine MRI relies on breath-hold Cartesian acquisitions, which are vulnerable to motion artifacts and can be uncomfortable or infeasible, particularly for pediatric and other noncompliant patients who cannot reliably hold their breath. Free-breathing radial acquisitions can alleviate these limitations, but robust reconstruction at high acceleration remains challenging due to prominent streak artifacts. To address these limitations, we propose Cine-DL, a clinically oriented framework that couples targeted k-space preprocessing with fast, model-based deep reconstruction. In this pipeline, raw free-breathing radial data undergo retrospective cardiac binning and respiratory gating to resolve cardiac phases and discard motion-corrupted spokes. We then introduce Streak Optimized Coil Compression (SOC), which explicitly preserves cardiac signals while suppressing peripheral interference that typically drives the streak artifacts. The resulting 2D+t cine series is reconstructed with an unrolled network that alternates a ResNet proximal operator with physics-based data consistency updates solved via conjugate gradient. We further employ a memory-efficient training strategy that reduces peak memory usage. We evaluate Cine-DL on free-breathing volunteer data against established baselines (k-t SENSE and iGRASP) and demonstrate clinical translation via hospital deployment on newly acquired patient data. Our experiments show that Cine-DL consistently improves quantitative metrics and visual fidelity, supporting a practical route toward routine, time-sensitive clinical adoption of free-breathing cine MRI.

Jonathan W. Siegel, Snir Hordan, Hannah Lawrence et al.

The universal approximation theorem establishes that neural networks can approximate any continuous function on a compact set. Later works in approximation theory provide quantitative approximation rates for ReLU networks on the class of $α$-Hölder functions $f: [0,1]^N \to \mathbb{R}$. The goal of this paper is to provide similar quantitative approximation results in the context of group equivariant learning, where the learned $α$-Hölder function is known to obey certain group symmetries. While there has been much interest in the literature in understanding the universal approximation properties of equivariant models, very few quantitative approximation results are known for equivariant models. In this paper, we bridge this gap by deriving quantitative approximation rates for several prominent group-equivariant and invariant architectures. The architectures that we consider include: the permutation-invariant Deep Sets architecture; the permutation-equivariant Sumformer and Transformer architectures; joint invariance to permutations and rigid motions using invariant networks based on frame averaging; and general bi-Lipschitz invariant models. Overall, we show that equally-sized ReLU MLPs and equivariant architectures are equally expressive over equivariant functions. Thus, hard-coding equivariance does not result in a loss of expressivity or approximation power in these models.

Ali Syed, Aditya Nambiar, Jonathan W. Siegel

In many practical applications it is important to build symmetries into neural network architectures. Consider the important case of permutation symmetry on point clouds consisting of $n$ points in $d$ dimensions. In this case the network learns a function on a set of $n$ points in $\mathbb{R}^d$, and a natural paradigm for constructing invariant networks is Janossy pooling, which generalizes the popular Deep Sets architecture. We study the universality of this approach, in particular the important question of how large the embedding dimension must be to guarantee universality of this architecture. Specifically, using a novel technique, we prove new lower bounds on the required size of this embedding dimension. For Deep Sets, this gives the correct minimal dimension up to a constant factor for all $d > 1$. For $k$-ary Janossy pooling, we prove the first non-trivial lower bound on the required embedding dimension when $k > 1$.