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We compare the Finite Element Method (FEM) simulation of a standard Partial Differential Equation thermal problem of a plate with a hole with a Neural Network (NN) simulation. The largest deviation from the true solution obtained from FEM…

The architecture of a neural network and the selection of its activation function are both fundamental to its performance. Equally vital is ensuring these two elements are well-matched, as their alignment is key to achieving effective…

Most research in quantum computing today is performed against simulations of quantum computers rather than true quantum computers. Simulating a quantum computer entails implementing all of the unitary operators corresponding to the quantum…

The development of novel materials in recent years has been accelerated greatly by the use of computational modelling techniques aimed at elucidating the complex physics controlling microstructure formation in materials, the properties of…

As neural network model sizes have dramatically increased, so has the interest in various techniques to reduce their parameter counts and accelerate their execution. An active area of research in this field is sparsity - encouraging zero…

We present HadaCore, a modified Fast Walsh-Hadamard Transform (FWHT) algorithm optimized for the Tensor Cores present in modern GPU hardware. HadaCore follows the recursive structure of the original FWHT algorithm, achieving the same…

Meshless methods approximate operators in a specific node as a weighted sum of values in its neighbours. Higher order approximations of derivatives provide more accurate solutions with better convergence characteristics, but they come at…

A new method for implementing the kinetic energy operator for real-space, grid-based electronic structure codes is developed. It is based on multi-order Adaptive Finite Differencing (AFD) and uses atomic pseudo orbitals produced by the…

In a recent paper we have suggested that the finite temperature density matrix can be computed efficiently by a combination of polynomial expansion and iterative inversion techniques. We present here significant improvements over this…

Convolutional neural networks excel in image recognition tasks, but this comes at the cost of high computational and memory complexity. To tackle this problem, [1] developed a tensor factorization framework to compress fully-connected…

We introduced and analyzed robust recovery-based a posteriori error estimators for various lower order finite element approximations to interface problems in [9, 10], where the recoveries of the flux and/or gradient are implicit (i.e.,…

Computationally efficient classification system architecture is proposed. It utilizes fast tensor-vector multiplication algorithm to apply linear operators upon input signals . The approach is applicable to wide variety of recognition…

Multi-exit network is a promising architecture for efficient model inference by sharing backbone networks and weights among multiple exits. However, the gradient conflict of the shared weights results in sub-optimal accuracy. This paper…

Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of…

Floating point multiplication is one of the crucial operations in many application domains such as image processing, signal processing etc. But every application requires different working features. Some need high precision, some need low…

Tensor operations dominate modern computational workloads, yet their further acceleration demands hardware platforms with greater parallelism. Although photonic computing provides a compelling route for parallel processing, fully exploiting…

High-performance deep learning depends on efficient tensor programs. In recent years, automatic tensor program optimization, also known as tensor compilation, has emerged as the primary approach to generating efficient tensor programs.…

We analyze adaptive mesh-refining algorithms for conforming finite element discretizations of certain non-linear second-order partial differential equations. We allow continuous polynomials of arbitrary, but fixed polynomial order. The…

The recent surge of interest in Deep Neural Networks (DNNs) has led to increasingly complex networks that tax computational and memory resources. Many DNNs presently use 16-bit or 32-bit floating point operations. Significant performance…

Neural operators are a new type of models that can map between function spaces, allowing trained models to emulate the solution operators of partial differential equations (PDEs). This paper proposes a multigrid Fourier neural operator…