Paper quantifies neural operators' efficiency for solving nonlinear parabolic PDEs.
problem Quantifying the efficiency of neural operators for solving nonlinear parabolic PDEs.
method Deriving approximation rates by transferring PDEs to integral equations and leveraging Picard's iteration.
result Neural operators can efficiently approximate solution operators of nonlinear PDEs without exponential complexity growth.
We introduce a data-based approach to estimating key quantities which arise in the study of nonlinear control systems and random nonlinear dynamical systems. Our approach hinges on the observation that much of the existing linear theory may be readily extended to nonlinear systems - with a reasonable expectation of suc…
Study approximates nonlinear functionals using deep ReLU networks.
problem Approximating nonlinear continuous functionals with neural networks.
method Constructs continuous piecewise linear interpolation under simple triangulation, analyzes rates of approximation.
result Established rates of approximation for functional deep ReLU networks.
Kernel approximation via nonlinear random feature maps is widely used in speeding up kernel machines. There are two main challenges for the conventional kernel approximation methods. First, before performing kernel approximation, a good kernel has to be chosen. Picking a good kernel is a very challenging problem in its…
The study extends convergence guarantees for nonlinear TD learning, focusing on ReLU networks and reversibility.
problem Understanding convergence of nonlinear TD learning with function approximators.
method Analyzing TD(0) dynamics through nonlinear ODEs, considering function approximator geometry and reversibility.
result Global convergence to the true value function for well-conditioned function approximators in reversible environments.
Novel autoencoder method approximates Koopman operator in low dimensions.
problem Challenges in approximating finite Koopman operators using data-driven methods.
method Mori-Zwanzig autoencoder (MZ-AE) for robust Koopman operator approximation.
result Improved predictive capability and robust long-term statistical performance.
Novel filter uses deep BSDE for nonlinear density approximation.
problem Nonlinear filtering problem.
method Bayesian filter based on deep BSDE and neural networks.
result Theoretical convergence rate confirmed in numerical examples.
New method for nonlinear filtering and smoothing using factor graphs.
problem Handling deterministic nonlinear transformations in factor graphs.
method Approximate Gaussian message passing rules for factor graphs with Markov property.
result Proposed nonlinear modified Bryson-Frazier smoother.
New method combines variational inference with particle filtering for nonlinear data.
problem Combining variational inference and Monte Carlo sampling for nonlinear data.
method Formulates gradient steepest descent method based on local optimal transport principles, embeds local mappings in RKHS, uses approximations to avoid adjoint evaluation.
result RKHS approximation is highly successful and superior to ensemble approximation for nonlinear observational operators.
A novel model uses ODE-based random features to model nonlinear dynamical systems.
problem Modeling highly nonlinear dynamical systems with uncertainty quantification.
method Compositions of physics-informed random features derived from ODEs, combined with deep Gaussian processes and approximate Bayesian inference.
result The model effectively captures nonlinear behavior in real-world multivariate time series data and achieves comparable performance to other models on benchmark tasks.
The purpose of this paper is to analyze and compute the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility which can be a function of the second derivative of the option price itself. A motivation for studying the nonlinear Black--Scholes equation with a nonlinear vola…
Explains linearizing a nonlinear connection on a pullback bundle.
problem Clarifying the geometric meaning of linearized connections.
method Fiberwise linear approximation of a vector bundle connection.
result Clarifies the geometric meaning of linearized connections.
Method solves complex optimization problems with high probability bounds.
problem Nonlinear equality constrained stochastic optimization problems.
method Step-search sequential quadratic programming method.
result High-probability bound on iteration complexity for first-order stationarity.
Extends importance sampling to nonlinear models using adjoint operators.
problem Lack of tools for identifying important data points in nonlinear models.
method Introduces adjoint operator for nonlinear maps, generalizes norm and leverage scores.
result Generalized scores provide approximation guarantees for nonlinear mappings.
New method stabilizes inputs to DNN for secure inference with LHE.
problem Incompatibility of LHE with nonlinear functions in DNN.
method Training with polynomial approximations and Min-Max normalization.
result Loss in prediction accuracy reduced to small values or eliminated.
New method solves robust matrix completion using nonlinear equations.
problem Recover low rank and sparse matrices from incomplete observations.
method Transforms problem into solving a system of nonlinear equations, then uses the alternative direction method.
result Algorithm converges linearly to the true solution under proper assumptions.
Temporal Difference Learning analysis under non-i.i.d. data and nonlinear approximation.
problem Finite-sample behavior of TD(0) under non-i.i.d. data and nonlinear approximation.
method High-probability, finite-sample analysis of vanilla TD(0) on polynomially mixing Markov data, assuming Holder continuity and bounded generalized gradients.
result Bounds on the convergence rate of TD(0) with high probability, matching known i.i.d. rates and holding even with nonstationary initialization.
Enhances RSCNs with hybrid regularization for nonlinear dynamics.
problem Modeling nonlinear dynamic systems with uncertainties.
method Recurrent stochastic configuration networks with hybrid regularization.
result The method outperforms other models in nonlinear system identification and industrial tasks.
Tensor completion method identifies nonlinear systems from input-output data.
problem Identifying nonlinear functions from input-output data pairs.
method Formulated as tensor completion problem with smoothness regularization and solved using block coordinate descent.
result Provable correct nonlinear system identification under certain conditions.
Optimizes sparse signal recovery using nonlinear approximations.
problem Recovering sparse stochastic signals efficiently.
method Probabilistic approach with linear and nonlinear estimators.
result Structured estimator outperforms linear in MSE.
New method uses compositions of ReLU neural networks for nonlinear approximation.
problem Approximating functions with minimum error using a limited number of terms.
method Proposes dictionaries of functions in the form of compositions, implemented using ReLU FNNs.
result Improves approximation rate by using compositions, especially for deep networks.
Richberg technique adapted for nonlinear subequations.
problem Approximating strictly subharmonic functions in F-potential theory. method Adapting Richberg technique to F-potential theory. result Local approximation to global approximation for subharmonic functions.
Proposes semi-random features for nonlinear function approximation.
problem Nonlinear function approximation in machine learning.
method Semi-random features as a middle ground between deep learning and kernel methods.
result Proves universal approximation and generalization for deep semi-random features.
In many compressive sensing problems today, the relationship between the measurements and the unknowns could be nonlinear. Traditional treatment of such nonlinear relationships have been to approximate the nonlinearity via a linear model and the subsequent un-modeled dynamics as noise. The ability to more accurately ch…
Paper develops a new state estimation method for nonlinear systems.
problem State estimation for nonlinear state-space models is intractable.
method Developed a variational inference approach based on Gaussian approximations.
result The method outperforms alternative Gaussian approaches in various examples.
Kernel-based Bayesian filter for nonlinear systems using infinite-dimensional operators.
problem Modeling and predicting nonlinear dynamical systems.
method Functional Bayesian perspective, reproducing kernel Hilbert space, Gaussian kernel.
result Effective approximation and accurate results for nonlinear systems.
Study on statistical inference for nonlinear stochastic approximation with Markovian data.
problem Statistical inference for nonlinear stochastic approximation algorithms with Markovian data.
method Established a functional central limit theorem for the partial-sum process of the target parameter estimate, providing asymptotic pivotal statistics for constructing confidence intervals.
result Valid and efficient asymptotic inference method for nonlinear stochastic approximation algorithms with Markovian data.
This paper presents an efficient Bayesian framework for solving nonlinear, high-dimensional model calibration problems. It is based on a Variational Bayesian formulation that aims at approximating the exact posterior by means of solving an optimization problem over an appropriately selected family of distributions. The…
Paper uses Koopman operator and Nyström method for efficient nonlinear control.
problem Control of nonlinear dynamical systems.
method Combines Koopman operator framework with Nyström approximation for kernel methods.
result Theoretical guarantees on the convergence rates of the approximated Riccati operator and regulator objective.
We investigate finite-time decoupled convergence in nonlinear two-time-scale stochastic approximation.
problem Achieving decoupled convergence in nonlinear two-time-scale stochastic approximation.
method Nested local linearity assumption, suitable step size selection, convergence analysis of matrix cross term, fourth-order moment convergence rates.
result Finite-time decoupled convergence rates can be achieved in nonlinear two-time-scale stochastic approximation with proper step size selection.
Lazy training and mean field regimes studied for TD learning with nonlinear function approximation.
problem Approximating value function for MRP with TD learning and nonlinear functions.
method Lazy training and mean field scaling of parameters analyzed for convergence.
result Lazy training leads to exponential convergence to local/global minimizers, while mean field scaling results in all fixed points being minimizers.
Analytical formula and Newton's method compared for nonlinear Black-Scholes equations.
problem Solving nonlinear Black-Scholes parabolic equations with market illiquidity and risk factors.
method Comparison of analytical approximation formula and Newton's method.
result Accuracy and time complexity of both methods compared using market data.
Paper improves image retrieval quality using nonlinear rank approximations.
problem Improving image retrieval quality in high-dimensional feature spaces.
method Computes normalized approximated ranks, converts to similarities, and uses them in a new loss function.
result Significant improvement in image retrieval quality on multiple datasets.
Paper presents a new backward deep BSDE method for solving nonlinear FBSDE problems.
problem Nonlinear Forward Backward Stochastic Differential Equations (FBSDE) with terminal conditions.
method Backward deep BSDE method applied to FBSDE with nonlinear generators and random initial conditions.
result Derives exact and Taylor-based approximations for time-stepping nonlinear BSDEs.
A deep learning method solves nonlinear filtering problems efficiently.
problem Nonlinear filtering problem
method Deep splitting method combined with energy-based neural network approximation
result Computational efficiency and performance comparable to Kalman and bootstrap filters
This paper is a follow up to the previous author's paper on convex optimization. In that paper we began the process of adjusting greedy-type algorithms from nonlinear approximation for finding sparse solutions of convex optimization problems. We modified there three the most popular in nonlinear approximation in Banach…
The purpose of this survey chapter is to present a transformation technique that can be used in analysis and numerical computation of the early exercise boundary for an American style of vanilla options that can be modelled by class of generalized Black-Scholes equations. We analyze qualitatively and quantitatively the…
The purpose of this paper is to construct the early exercise boundary for a class of nonlinear Black--Scholes equations with a nonlinear volatility depending on the option price. We review a method how to transform the problem into a solution of a time depending nonlinear parabolic equation defined on a fixed domain. R…
Method approximates efficient frontier of chance-constrained programs.
problem Approximating the efficient frontier of chance-constrained nonlinear programs.
method Stochastic approximation method based on bi-objective viewpoint.
result Converges to stationary solutions of a smooth approximation of the original problem.
Paper analyzes and proves convergence of a new method for solving complex PDEs.
problem Solving high-dimensional nonlinear PDEs and PIDEs with random neural networks.
method Random deep splitting method using random neural networks.
result The method converges to the unique viscosity solution of nonlinear PDEs and PIDEs.
Improved bounds for function approximation in nonlinear sets.
problem Achieving high probability error with limited samples in nonlinear function approximation.
method Restricting model class to a neighbourhood of the best approximation and estimating sample complexity using tangent and normal spaces' complexities and curvature.
result Improved worst-case bounds for sample complexity in more general sets like tensor networks and neural networks.
Families of explicit solutions are found to a nonlinear Black-Scholes equation which incorporates the feedback-effect of a large trader in case of market illiquidity. The typical solution of these families will have a payoff which approximates a strangle. These solutions were used to test numerical schemes for solving …
Study perpetual put options using nonlinear Black-Scholes equations.
problem Analyzing early exercise boundaries for perpetual put options.
method Transformed into a nonlinear stationary Black-Scholes equation and solved numerically.
result Numerical results of early exercise boundary, option price and their parameters.
Develops a framework for learning nonlinear operators using Mercer kernels.
problem Learning nonlinear operators between infinite-dimensional spaces.
method Stochastic approximation framework with Mercer operator-valued kernels.
result Establishes dimension-free polynomial convergence rates for nonlinear operator learning.
LaRP framework improves object classification using random projections.
problem Efficiently approximating nonlinear kernels in high-dimensional spaces.
method Separates linear kernels and nonlinearity using a layered random projection approach.
result Notable improvement in object classification performance.
FNOs learn solution operators of dissipative equations efficiently via spectral methods.
problem Learning and approximation of solution operators for dissipative equations.
method Introducing spectral methods and deriving FNO approximation bounds and sample complexity guarantees.
result Polynomial sample complexity guarantees for FNOs learning solution operators of dissipative equations.
Sign α-stable random projections approximate nonlinear kernels for large-scale learning.
problem Efficiently approximating nonlinear kernels for large-scale machine learning.
method Sign α-stable random projections for data processing.
result Approximation of various nonlinear kernels depending on α.
New method solves high-dimensional PDEs and 2BSDEs efficiently.
problem High-dimensional fully nonlinear PDEs and 2BSDEs in financial models.
method Connection between PDEs and 2BSDEs, merged formulation, temporal discretization, spatial approximation via neural nets, stochastic gradient descent.
result Efficient and accurate solution for high-dimensional nonlinear expectations.