arXiv research

Upper bounds on neural network complexity for PDE solutions.

Minimum-norm solutions generalize well in over-parametrized neural networks.

Researchers found solutions to minimal surface equations in 6D.

A new parametric method studies Willmore flows and energy quantization.

X-TFC solves parametric DEs with neural networks and physics constraints.

High dimensional sparse learning has imposed a great computational challenge to large scale data analysis. In this paper, we are interested in a broad class of sparse learning approaches formulated as linear programs parametrized by a {\em regularization factor}, and solve them by the parametric simplex method (PSM). O…

We solve the mean parametrization of von Mises-Fisher distribution.

A new principle minimizes residual and introduces momentum to improve PDE solution dynamics.

Bayesian approach for solving systems of linear PDEs with boundary conditions.

Optimizes fairness without sacrificing primary objectives.

Efficiently models event-based data with general parametric kernels.

Error estimates for nonlinear PDEs using kernel/GP methods.

We give a Lie-theoretic explanation for the convex polytope which parametrizes the globally smooth solutions of the topological-antitopological fusion equations of Toda type (tt$^*$-Toda equations) which were introduced by Cecotti and Vafa. It is known from [GL] [GIL1] [M1] [M2] that these solutions can be parametrized…

Deep neural network approximates multivariate option pricing.

Study uses neural networks to solve complex equations efficiently.

Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determ…

We show that Scherk's first surface, a one-parameter family of solutions to the minimal surface equation, may be written as a linear superposition of other solutions with specific parametric values.

GOL uses semi-parametric approach to learn from single examples in autonomous driving.

Over-parametrization speeds up learning a single neuron model.

We provide new theoretical insights on why over-parametrization is effective in learning neural networks. For a $k$ hidden node shallow network with quadratic activation and $n$ training data points, we show as long as $k \ge \sqrt{2n}$, over-parametrization enables local search algorithms to find a \emph{globally} op…

Deep learning solves high-dimensional PDEs efficiently.

Extends Demographic Parity for fairer wage predictions with expert knowledge.

Maximal causal curves for Lipschitz metrics are either lightlike or timelike.

DeepAveragers solves offline RL by solving derived MDPs from static data.

New method solves high-dimensional Bayesian inverse problems efficiently.

SMD outperforms SGD in over-parametrized linear models for certain data distributions.

Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.

Enhances selective inference for generalized lasso using parametric programming.

We outline a new geometric method of constructing exact solutions of gravitational field equations parametrized by generic off-diagonal metrics, anholonomic frames and possessing, in general, nontrivial torsion and nonmetricity. The formalism of nonlinear connections is elaborated for (pseudo) Riemannian and Einstein-C…

Meta-learning base distributions for efficient PDE solutions.

We introduce performance-based regularization (PBR), a new approach to addressing estimation risk in data-driven optimization, to mean-CVaR portfolio optimization. We assume the available log-return data is iid, and detail the approach for two cases: nonparametric and parametric (the log-return distribution belongs in …

We introduce a balloon estimator in a generalized expectation-maximization method for estimating all parameters of a Gaussian mixture model given one data sample per mixture component. Instead of limiting explicitly the model size, this regularization strategy yields low-complexity sparse models where the number of eff…

The aim of this paper is to construct and analyze solutions to a class of Hamilton-Jacobi-Bellman equations with range bounds on the optimal response variable. Using the Riccati transformation we derive and analyze a fully nonlinear parabolic partial differential equation for the optimal response function. We construct…

Differentiable cutting-plane layers solve parametric mixed-integer linear optimization problems.

Develops a simple method for creating private confidence intervals.

ADA augments data using AR replicas for robust regression.

We consider grouping as a general characterization for problems such as clustering, community detection in networks, and multiple parametric model estimation. We are interested in merging solutions from different grouping algorithms, distilling all their good qualities into a consensus solution. In this paper, we propo…

NPOD algorithm improves efficiency in estimating pharmacokinetic parameters.

In this paper, for the Lorentz manifold $M^{2}\times\mathbb{R}$, with $M^{2}$ a $2$-dimensional complete surface with nonnegative Gaussian curvature, we investigate its space-like graphs over compact strictly convex domains in $M^{2}$, which are evolving by the non-parametric mean curvature flow with prescribed contact…

We present a non-parametric Bayesian approach to structure learning with hidden causes. Previous Bayesian treatments of this problem define a prior over the number of hidden causes and use algorithms such as reversible jump Markov chain Monte Carlo to move between solutions. In contrast, we assume that the number of hi…

We consider strictly convex hypersurfaces which are evolving by the non-parametric logarithmic Gauss curvature flow subject to a Neumann boundary condition. Solutions are shown to converge smoothly to hypersurfaces moving by translation. In particular, for bounded domains we prove that convex functions with prescribed …

We develop several deep learning algorithms for approximating families of parametric PDE solutions. The proposed algorithms approximate solutions together with their gradients, which in the context of mathematical finance means that the derivative prices and hedging strategies are computed simulatenously. Having approx…

Paper analyzes RTSE on AE manifolds and closed manifolds, showing well-posedness and parametrization of solutions.

We prove that conformally parametrized surfaces in Euclidean space $\Rcubec$ of curvature $c$ admit a symmetry reduction of their Gauss-Codazzi equations whose general solution is expressed with the sixth Painlevé function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko, Eitner an…

In recent work on both generative and discriminative score to log-likelihood-ratio calibration, it was shown that linear transforms give good accuracy only for a limited range of operating points. Moreover, these methods required tailoring of the calibration training objective functions in order to target the desired r…

We propose kernel-based collocation methods for numerical solutions to Heath-Jarrow-Morton models with Musiela parametrization. The methods can be seen as the Euler-Maruyama approximation of some finite dimensional stochastic differential equations, and allow us to compute the derivative prices by the usual Monte Carlo…

In this paper, the Weierstrass technique for harmonic maps S^2 -> CP^(N-1) is employed in order to obtain surfaces immersed in multidimensional Euclidean spaces. It is shown that if the CP^(N-1) model equations are defined on the sphere S^2 and the associated action functional of this model is finite, then the generali…

We find exact solutions describing Ricci flows of four dimensional pp-waves nonlinearly deformed by two/three dimensional solitons. Such solutions are parametrized by five dimensional metrics with generic off-diagonal terms and connections with nontrivial torsion which can be related, for instance, to antisymmetric ten…