Related papers: Feature Engineering with Regularity Structures
In many real world applications of machine learning, models have to meet certain domain-based requirements that can be expressed as constraints (e.g., safety-critical constraints in autonomous driving systems). Such constraints are often…
We study model spaces, in the sense of Hairer, for stochastic partial differential equations involving the fractional Laplacian. We prove that the fractional Laplacian is a singular kernel suitable to apply the theory of regularity…
This work proposes a machine-learning framework for constructing statistical models of errors incurred by approximate solutions to parameterized systems of nonlinear equations. These approximate solutions may arise from early termination of…
Regularization techniques are widely employed in optimization-based approaches for solving ill-posed inverse problems in data analysis and scientific computing. These methods are based on augmenting the objective with a penalty function,…
We present a methodology combining neural networks with physical principle constraints in the form of partial differential equations (PDEs). The approach allows to train neural networks while respecting the PDEs as a strong constraint in…
Physics Informed Neural Networks is a numerical method which uses neural networks to approximate solutions of partial differential equations. It has received a lot of attention and is currently used in numerous physical and engineering…
We give a novel characterization of the centered model in regularity structures which persists for rough drivers even as a mollification fades away. We present our result for a class of quasilinear equations driven by noise, however we…
This work proposes an autoencoder neural network as a non-linear generalization of projection-based methods for solving Partial Differential Equations (PDEs). The proposed deep learning architecture presented is capable of generating the…
In this work we are interested in the problems of supervised learning and variable selection when the input-output dependence is described by a nonlinear function depending on a few variables. Our goal is to consider a sparse nonparametric…
Stochastic differential equations (SDEs) are a ubiquitous modeling framework that finds applications in physics, biology, engineering, social science, and finance. Due to the availability of large-scale data sets, there is growing interest…
We propose semi-random features for nonlinear function approximation. The flexibility of semi-random feature lies between the fully adjustable units in deep learning and the random features used in kernel methods. For one hidden layer…
The features in high dimensional biomedical prediction problems are often well described with lower dimensional manifolds. An example is genes that are organised in smaller functional networks. The outcome can then be described with the…
A machine learning (ML) feature network is a graph that connects ML features in learning tasks based on their similarity. This network representation allows us to view feature vectors as functions on the network. By leveraging function…
This work proposes a mathematical approach that (re)defines a property of Machine Learning models named stability and determines sufficient conditions to validate it. Machine Learning models are represented as functions, and the…
Partial differential equations (PDEs) govern a wide range of physical phenomena, but their numerical solution remains computationally demanding, especially when repeated simulations are required across many parameter settings. Recent…
Our goal is to provide a review of deep learning methods which provide insight into structured high-dimensional data. Rather than using shallow additive architectures common to most statistical models, deep learning uses layers of…
A time series represents a set of observations collected over time. Typically, these observations are captured with a uniform sampling frequency (e.g. daily). When data points are observed in uneven time intervals the time series is…
Nested Effects Models (NEMs) are a class of graphical models introduced to analyze the results of gene perturbation screens. NEMs explore noisy subset relations between the high-dimensional outputs of phenotyping studies, e.g. the effects…
We propose and analyze a regularization approach for structured prediction problems. We characterize a large class of loss functions that allows to naturally embed structured outputs in a linear space. We exploit this fact to design…
In model selection problems for machine learning, the desire for a well-performing model with meaningful structure is typically expressed through a regularized optimization problem. In many scenarios, however, the meaningful structure is…