arXiv research

INO learns physical models with momentum conservation laws.

Frame Averaging makes neural networks invariant or equivariant to new symmetries.

Physics-informed neural networks improve by measuring effective dimensionality of constraints.

This research studies affine invariance in continuous-domain convolutional neural networks.

A universal collection of 4 invariants improves neural network accuracy for molecular dynamics.

Transformationally invariant processors constructed by transformed input vectors or operators have been suggested and applied to many applications. In this study, transformationally identical processing based on combining results of all sub-processes with corresponding transformations at one of the processing steps or …

MCNO learns PDE solution operators using Monte Carlo sampling.

GeONet learns the Wasserstein geodesic without mesh discretization.

ISOKANN learns collective variables and effective dynamics for metastable transitions.

VANO uses neural operators for unsupervised learning of functional data.

In this paper we propose a generalization of deep neural networks called deep function machines (DFMs). DFMs act on vector spaces of arbitrary (possibly infinite) dimension and we show that a family of DFMs are invariant to the dimension of input data; that is, the parameterization of the model does not directly hinge …

This paper completes the construction of arbitrary order conformally invariant differential operators in higher spin spaces. Jan Slovák has classified all conformally invariant differential operators on locally conformally flat manifolds. We complete his results in higher spin theory by giving explicit expressions for …

New method learns PDE solutions from low-fidelity data.

A new graph neural network framework captures long-range interactions efficiently.

Convolutional Neural Networks (CNNs) have demonstrated state-of-the-art performance on many visual recognition tasks. However, the combination of convolution and pooling operations only shows invariance to small local location changes in meaningful objects in input. Sometimes, such networks are trained using data augme…

Graph Neural Networks (GNN) come in many flavors, but should always be either invariant (permutation of the nodes of the input graph does not affect the output) or equivariant (permutation of the input permutes the output). In this paper, we consider a specific class of invariant and equivariant networks, for which we …

Enhanced Yang-Baxter operators give rise to invariants of oriented links. We expand the enhancing method to generalized Yang-Baxter operators. At present two examples of generalized Yang-Baxter operators are known and recently three types of variations for one of these were discovered. We present the definition of enha…

The paper examines scalar fourth-order linear differential operators and their invariants.

The low displacement rank (LDR) framework for structured matrices represents a matrix through two displacement operators and a low-rank residual. Existing use of LDR matrices in deep learning has applied fixed displacement operators encoding forms of shift invariance akin to convolutions. We introduce a class of LDR ma…

Study natural invariants for differential operators, simplifying their equivalence problem.

The paper studies differential operator invariants and equivalence under Lie pseudogroups.

It is well known that neural networks with rectified linear units (ReLU) activation functions are positively scale-invariant. Conventional algorithms like stochastic gradient descent optimize the neural networks in the vector space of weights, which is, however, not positively scale-invariant. This mismatch may lead to…

Study natural invariants for third order nonlinear operators on 2D manifolds.

The paper classifies invariant operators and proves a Liouville theorem.

Enhances GNNs by capturing node relationships, outperforming 2-WL test.

Study presents a method to induce a generalized neural network from joint group invariant functions.

Recently, researchers have started applying convolutional neural networks (CNNs) with one-dimensional convolutions to clinical tasks involving time-series data. This is due, in part, to their computational efficiency, relative to recurrent neural networks and their ability to efficiently exploit certain temporal invari…

The study quantifies how many objects can be linearly classified under all views.

Study invariant operators and vanishing theorems in CR geometry.

New calculus for pseudodifferential operators on manifolds with cylindrical ends.

Neural networks learn from ensemble forecasts without considering their order.

In this article, we investigate differential operators on the Siegel-Jacobi space that are invariant under the natural action of the Jacobi group. These invariant differential operators play an important role in the arithmetic theory of Jacobi forms of higher degree. We present some explicit invariant differential oper…

Study invariant operations on Fedosov manifolds.

New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalisations as well as 4-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all invariants and invariant operators arise from these universal operators and that the…

BGG-operators form sequences of invariant differential operators and the first of these is overdetermined. Interesting equations in conformal geometry described by these operators are those for Einstein scales, conformal Killing forms and conformal Killing tensors. We present a deformation procedure of the tractor conn…

Study Schiffer operators on Riemann surfaces, linking conformal and topological invariants.

We seek to improve deep neural networks by generalizing the pooling operations that play a central role in current architectures. We pursue a careful exploration of approaches to allow pooling to learn and to adapt to complex and variable patterns. The two primary directions lie in (1) learning a pooling function via (…

The paper studies chambered invariants of real Cauchy-Riemann operators on Riemann surfaces.

Multiple instance learning (MIL) is a variation of supervised learning where a single class label is assigned to a bag of instances. In this paper, we state the MIL problem as learning the Bernoulli distribution of the bag label where the bag label probability is fully parameterized by neural networks. Furthermore, we …

We consider two approaches to isotopy invariants of oriented links: one from ribbon categories and the other from generalized Yang-Baxter operators with appropriate enhancements. The generalized Yang-Baxter operators we consider are obtained from so-called gYBE objects following a procedure of Kitaev and Wang. We show …

The paper addresses instability in CNNs' first layer by proving max pooling's shift invariance.

Proves formal self-adjointness of certain differential operators.

Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.

Classifies curved bidifferential operators on manifolds.

We identify Melrose's suspended algebra of pseudodifferential operators with a subalgebra of the algebra of parametric pseudodifferential operators with parameter space $\R$. For a general algebra of parametric pseudodifferential operators, where the parameter space may now be a cone $Γ\subset\R^p$, we construct a uniq…

Study how Turaev-Viro invariants change with cabling operations.

This paper introduces CloudLSTM, a new branch of recurrent neural models tailored to forecasting over data streams generated by geospatial point-cloud sources. We design a Dynamic Point-cloud Convolution (DConv) operator as the core component of CloudLSTMs, which performs convolution directly over point-clouds and extr…

In this note, we consider the problem of constructing knot invariants from Yang-Baxter operators associated to (unitary associative) algebra structures. We first compute the enhancements of these operators. Then, we conclude that Turaev's procedure to construct knot invariants, as modified by Murakami, invariably produ…