Related papers: Projection methods for Neural Field equations
Over the past two decades, mobile imaging has experienced a profound transformation, with cell phones rapidly eclipsing all other forms of digital photography in popularity. Today's cell phones are equipped with a diverse range of imaging…
Nowadays, neural networks are widely used in many applications as artificial intelligence models for learning tasks. Since typically neural networks process a very large amount of data, it is convenient to formulate them within the…
Random projections offer an appealing and flexible approach to a wide range of large-scale statistical problems. They are particularly useful in high-dimensional settings, where we have many covariates recorded for each observation. In…
Self-awareness is the key capability of autonomous systems, e.g., autonomous driving network, which relies on highly efficient time series forecasting algorithm to enable the system to reason about the future state of the environment, as…
Several variants of Neural Radiance Fields (NeRFs) have significantly improved the accuracy of synthesized images and surface reconstruction of 3D scenes/objects. In all of these methods, a key characteristic is that none can train the…
Deep learning-based models have demonstrated remarkable success in solving illposed inverse problems; however, many fail to strictly adhere to the physical constraints imposed by the measurement process. In this work, we introduce a…
Stochastic Gradient Descent (SGD) has proven to be remarkably effective in optimizing deep neural networks that employ ever-larger numbers of parameters. Yet, improving the efficiency of large-scale optimization remains a vital and highly…
We present a numerical framework for solving neural field equations on surfaces using Radial Basis Function (RBF) interpolation and quadrature. Neural field models describe the evolution of macroscopic brain activity, but modeling studies…
Multilevel methods are among the most efficient numerical methods for solving large-scale linear systems that arise from discretized partial differential equations. The fundamental module of such methods is a two-level procedure, which…
A general high-order fully explicit scheme based on projective integration methods is here presented to solve systems of degenerate parabolic equations in general dimensions. The method is based on a BGK approximation of the…
With the rise of neural models across the field of information retrieval, numerous publications have incrementally pushed the envelope of performance for a multitude of IR tasks. However, these networks often sample data in random order,…
Camera placement is crutial in multi-camera systems such as virtual reality, autonomous driving, and high-quality reconstruction. The camera placement challenge lies in the nonlinear nature of high-dimensional parameters and the…
We deal with the problem of bridging the gap between two scales in neuronal modeling. At the first (microscopic) scale, neurons are considered individually and their behavior described by stochastic differential equations that govern the…
In this paper we study linear projection methods for approximating the solution and simultaneously preserving first integrals of autonomous ordinary differential equations. We show that (linear) projection methods are a subset of discrete…
Differential equations are a ubiquitous tool to study dynamics, ranging from physical systems to complex systems, where a large number of agents interact through a graph with non-trivial topological features. Data-driven approximations of…
We proposed a framework for solving inverse problems in differential equations based on neural networks and automatic differentiation. Neural networks are used to approximate hidden fields. We analyze the source of errors in the framework…
We introduce and analyze a discontinuous Galerkin method for the numerical modelling of the equations of Multiple-Network Poroelastic Theory (MPET) in the dynamic formulation. The MPET model can comprehensively describe functional changes…
The accuracy of solving partial differential equations (PDEs) on coarse grids is greatly affected by the choice of discretization schemes. In this work, we propose to learn time integration schemes based on neural networks which satisfy…
Unrolling training trajectories over time strongly influences the inference accuracy of neural network-augmented physics simulators. We analyze this in three variants of training neural time-steppers. In addition to one-step setups and…
This work begins by establishing a mathematical formalization between different geometrical interpretations of Neural Networks, providing a first contribution. From this starting point, a new interpretation is explored, using the idea of…