Method uses neural networks to fit nonlinear operators from data.
problem Finding nonlinear integro-differential operators from data.
method Parametrizes spatial operator with neural networks and Fourier transforms.
result Can recover spatial operators in fractional heat and Kuramoto-Sivashinsky equations.
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.
Study natural invariants for third order nonlinear operators on 2D manifolds.
problem Equivalence problem of third order nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to equivalence problem.
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.
Develops a Krylov subspace method for estimating nonlinear systems with random noise.
problem Estimating nonlinear dynamical systems with random noise.
method Lifted representation of nonlinear dynamical systems using transfer operators, extended Arnoldi method, and shift-invert Arnoldi method.
result Empirical validation of methods on synthetic and real-world healthcare data.
The paper tackles data-driven optimal control of unknown nonlinear systems using RKHS.
problem Unknown nonlinear dynamics and stage cost functions.
method Embed state densities into RKHS, learn Markov operators, solve Hamilton-Jacobi-Bellman recursions.
result Solves a wide range of nonlinear control problems, including depth regulation.
AdaKoop efficiently models nonlinear dynamics from nonstationary data streams.
problem Capturing nonlinear dynamics in nonstationary data streams with computational efficiency.
method Koopman operator theory and probabilistic framework for streaming data.
result AdaKoop outperforms state-of-the-art methods in real-time forecasting accuracy and efficiency.
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.
Kernel method approximates Koopman operator eigenfunctions.
problem Complexity of computing Koopman operator spectra.
method Kernel-based approach to construct principal eigenfunctions.
result Principal eigenfunctions match linearization eigenvalues.
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.
Study on maximum principles for nonlinear equations on Riemannian manifolds.
problem Investigating strong maximum principles for fully nonlinear equations on Riemannian manifolds.
method Analyzing scaling conditions and applying to various nonlinear operators.
result Established new strong comparison principles for second order uniformly elliptic problems.
Finslerian graph neural networks recover nonlinear diffusion geometry
problem Graph neural networks on point clouds
method Estimates of the Finsler Laplacian
result Recovery of Finsler geometry
Study natural invariants for differential operators, simplifying their equivalence problem.
problem Equivalence problem of nonlinear differential operators.
method Description of rational natural differential invariants.
result Application of natural invariants to simplify differential operator equivalence.
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.
Study generalizes Picard iteration for nonlinear PDEs, deriving bounds on error.
problem Generalize Picard iteration for nonlinear parabolic PDEs.
method Formulate Picard iteration as abstract state-transition model, derive generalization error bounds.
result Picard depth reduction reduces Picard truncation error without increasing estimation error.
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.
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.
New method identifies key genes affecting phenotypes in biological systems.
problem Identifying genes that drive specific phenotypes in complex biological systems.
method Data-driven observability decomposition using Koopman operators.
result Koopman operator representation identifies genes that drive phenotypes.
New algorithm learns Koopman operator online, with complexity control and convergence guarantees.
problem Online learning of Koopman operator for general nonlinear systems.
method Sparse online learning via stochastic approximation, RKHS action, CME operator.
result Provably convergent algorithm with finite-time guarantees in mis-specified setting.
Unified bounds for neural networks incorporating physical laws.
problem Limitations in existing generalization analyses for PINNs and VPINNs.
method Unified framework using Taylor expansion and Koopman-based analysis.
result High-rank networks can generalize well even with differential operators.
Developed a BP algorithm for training neural networks with 2nd order neurons.
problem Training neural networks with nonlinear quadratic operations.
method Created a general backpropagation algorithm.
result Validated the generalized BP algorithm through numerical studies.
Solves curvature problems on manifolds with negative curvature.
problem Prescribed curvature problems on closed manifolds with negative curvature.
method Investigates fully nonlinear prescribed curvature problems for modified Schouten tensor on closed Riemannian manifolds with negative curvature.
result Proves solvability of curvature problems under certain conditions.
This paper studies Loday algebroids, introducing new concepts and formulas.
problem Exploring Loday algebroids and their cohomology, nonlinear connections, and characteristic classes.
method Introducing action Loday algebroids, clarifying Loday algebroid morphisms, studying nonlinear connections, and defining secondary characteristic classes.
result Established a generalized Chern-Simons formula for nonlinear connections on Loday algebroids.
Develops a metric for comparing nonlinear dynamical systems.
problem Developing a metric for nonlinear dynamical systems.
method Using Perron-Frobenius operators in reproducing kernel Hilbert spaces.
result Includes existing fundamental metrics as special cases.
Researchers develop a method to control nonlinear systems with Koopman operator regression.
problem Controlling nonlinear systems with finite action spaces.
method Koopman operator regression for dynamics estimation and model predictive control for control.
result The method yields a linear switching predictive model for control.
Novel neural operator predicts complex spatiotemporal dynamics from partial observations.
problem Capturing complex operator dynamics in infinite-dimensional function spaces.
method Integrates Koopman operator theory with deep neural networks to approximate nonlinear operators between Banach spaces.
result BNO achieves robust zero-shot super-resolution in unsteady flow prediction and outperforms conventional methods.
The paper studies boundedness of pseudo-differential operators on smooth manifolds.
problem Boundedness of pseudo-differential operators in Lp-Lq spaces on smooth manifolds. method Using global symbols and extending Hörmander's condition, the paper investigates Lp-boundedness, L∞-BMO estimates, and Lp-Lq boundedness for Fourier multipliers and pseudo-differential operators. result The paper proves Lp-Lq boundedness for the range 1<p≤2≤q<∞. Properties of the Cauchy-Riemann-Fueter equation for maps between quaternionic manifolds are studied. Spaces of solutions in case of maps from a K3-surface to the cotangent bundle of a complex projective space are computed. A relationship between harmonic spinors of a generalized nonlinear Dirac operator and solutions …
Neural operators correct PDE residuals to improve BIP solutions.
problem Reducing error in infinite-dimensional Bayesian inverse problems with neural operators.
method Error correction using PDE residuals to improve neural operator approximation.
result Trained neural operators with error correction achieve a quadratic reduction in approximation error.
Proposes a Koopman operator method for time-dependent reliability analysis of nonlinear systems.
problem Challenges in time-dependent reliability analysis of nonlinear dynamical systems.
method Koopman operator approach for transforming nonlinear systems into linear ones, combined with deep learning for intrinsic coordinates.
result Robust and generalizable approach for time-dependent reliability analysis, superior to purely data-driven methods.
DeepONet learns nonlinear operators from data to identify differential equations.
problem Learning nonlinear operators from data to identify differential equations.
method DeepONet architecture with branch and trunk nets.
result DeepONet significantly reduces generalization error compared to fully-connected networks.
Improved VQA accuracy with generalized fusion operators.
problem Enhancing multimodal fusion for better VQA performance.
method Generalized Hadamard-Product fusion operators with Nonlinearity Ensembling, Feature Gating, and post-fusion layers.
result 1.1% absolute improvement on VQA 2.0 test-dev set.
Deep learning discovers intrinsic coordinates for nonlinear dynamics.
problem Identifying coordinate transformations for strongly nonlinear dynamics.
method Modified auto-encoder network to discover Koopman eigenfunctions.
result Compact and efficient embedding of intrinsic rank for nonlinear systems.
Data-driven control of robotic systems using Koopman operators with error bounds.
problem Real-time control of nonlinear robotic systems with unknown dynamics.
method Constructing a Koopman operator-based linear representation using higher-order derivatives of nonlinear dynamics, with error bounds derived from Taylor series accuracy analysis.
result The Koopman model provides marginally better performance than competing nonlinear modeling methods and can be efficiently controlled using linear control design tools.
Solves nonlinear problems on metric structures through eigenvalue counting.
problem Nonlinear equations on metric structures
method Counting large eigenvalues of linearized operators
result Solves fully nonlinear Loewner-Nirenberg and Yamabe problems
Study uses SGD to learn operators in Hilbert spaces with convergence analysis.
problem Learning operators in general Hilbert spaces with SGD.
method Proposes weak and strong regularity conditions for convergence analysis.
result SGD converges to best linear approximation of nonlinear operators.
The paper proposes a new method for creating interpretable models using convex optimization.
problem Creating models that are both accurate and interpretable for decision-making.
method Formulates convex learning problems that combine interpretability with accuracy, using operator theory and parametric nonlinear models.
result Shows how to create efficient surrogate models that are both accurate and interpretable.
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.
The paper examines geometric properties of domains for the p-Laplacian in Euclidean and hyperbolic spaces.
problem Exploring geometric properties of unbounded extremal domains for the p-Laplacian operator.
method Analyzing properties in Euclidean and hyperbolic spaces, proving constraints on domains and their asymptotic boundaries.
result Extremal domains in two dimensions must be balls, and in hyperbolic space, they have specific geometric constraints.
We give an explicit formula, as a formal differential operator, for quantum microformal morphisms of (super)manifolds that we introduced earlier. Such quantum microformal morphisms are essentially oscillatory integral operators or Fourier integral operators of a particular kind. They act on oscillatory wave functions, …
Proposes new nonlinear scoring functions to maximize pAUC in binary classification.
problem Inaccurate binary classification measures in biased prior probability settings.
method Introduces nonlinear scoring functions based on generative models and deep neural networks.
result Nonlinear scoring functions improve pAUC maximization compared to linear functions.
New integrable systems are created using matrix operations and Lie algebra elements.
problem Generating coupled nonlinear integrable systems from zero curvature equation.
method Constructing Maurer-Cartan forms using Kronecker product and specific matrices (nilpotent, Hadamard, idempotent, k-idempotent).
result Found a closure property among chosen matrices crucial for coupling and nonlinearity.
Model proposes neural network for continuous time dynamics with inductive biases.
problem Training neural networks for small datasets with nonlinear dynamics.
method Inductive biases on decay rates and frequencies using Koopman operator theory.
result Higher forecasting performance with single short training sequence.
Survey on a nonlinear d'Alembertian from general relativity.
problem Understanding a new nonlinear operator from relativity.
method Review of a new distributional d'Alembertian, comparison estimates, and exact representation formulas.
result Control of the timelike cut locus through optimal transport.
The paper concerns singular solutions of nonlinear elliptic equations, which include removable singularities for viscosity solutions, a strengthening of the Hopf Lemma including parabolic equations, Strong maximum principle and Hopf Lemma for viscosity solutions including also parabolic equations.
Study shows neural operators can efficiently solve complex reaction-diffusion systems.
problem Efficiently solving nonlinear reaction-diffusion systems using neural operators.
method Laplacian-based neural operators applied to a generalized Gierer-Meinhardt system.
result Explicit approximation error bounds established for neural operators in terms of network parameters.
Proposes a new method for nonlinear Bayesian updates using ensemble kernel regression.
problem Nonlinear and non-Gaussian Bayesian updates for complex systems.
method Combines Kalman filtering for observed components and kernel density estimation for unobserved components, with subsampling and clustering.
result Reduces estimation errors in highly nonlinear scenarios compared to standard linear updates.
Interdisciplinary study linking potential theory and elliptic PDEs.
problem Understanding solutions to nonlinear elliptic PDEs.
method Combining geometric and potential theory approaches.
result Validity of comparison principle and existence/uniqueness of solutions.