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.
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.
This note introduces a regression technique for finding a class of nonlinear integro-differential operators from data. The method parametrizes the spatial operator with neural networks and Fourier transforms such that it can fit a class of nonlinear operators without needing a library of a priori selected operators. We…
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<∞. 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.
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.
Bayesian model learns physics laws from data with uncertainty quantification.
problem Lack of uncertainty in discovering governing physical laws from data.
method Bayesian approach with leaf and root modules, Gaussian process for operators, automatic differentiation.
result Quantifies reliability of learned physics laws and propagates uncertainty.
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.
Uniform estimates for complex equations on compact manifolds found.
problem Uniform estimates for (n−1)−form fully nonlinear PDEs on compact Hermitian manifolds. method Local comparison with Monge-Ampère equations and finding an appropriate elliptic operator.
result A priori L∞ estimate for the equations. New method deflates manifolds to visualize high-dimensional data.
problem Failure of nonlinear dimensionality reduction methods on simple manifolds.
method Iterative deflation of differential operators using single-coordinate estimates.
result Empirically, recovers novel embeddings on real-world and synthetic datasets.
Bayesian methods solve complex nonlinear PDEs efficiently.
problem Solving nonlinear PDEs with high computational cost.
method Bayesian inference with approximate likelihood based on discretization.
result Probabilistic uncertainty quantification for PDE solutions is feasible.
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.
In this paper, we present an initial attempt to learn evolution PDEs from data. Inspired by the latest development of neural network designs in deep learning, we propose a new feed-forward deep network, called PDE-Net, to fulfill two objectives at the same time: to accurately predict dynamics of complex systems and to …
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, …
Study solves optimal portfolio selection using HJB equation.
problem Optimal portfolio selection problem.
method Maximal monotone operator method, Banach fixed-point theorem, Fourier transform, monotone operators technique.
result Existence and uniqueness of solution to HJB equation.
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 Spencer complexes for Lie groupoids developed.
problem Developing Spencer complexes for Lie groupoids.
method Extending Malgrange's diagonal calculus to IimesG. result Construction of non-linear and linear Spencer complexes.
While it is widely known that neural networks are universal approximators of continuous functions, a less known and perhaps more powerful result is that a neural network with a single hidden layer can approximate accurately any nonlinear continuous operator. This universal approximation theorem is suggestive of the pot…
In this paper, we consider nonlinear PDEs in a port-Hamiltonian setting based on an underlying jet-bundle structure. We restrict ourselves to systems with 1-dimensional spatial domain and 2nd-order Hamiltonian including certain dissipation models that can be incorporated in the port- Hamiltonian framework by means of a…
We prove certain localized and global differential Harnack inequality for all positive solutions to the geometric conjugate heat equation coupled to the forward in time Ricci flow. In this case, the diffusion operator is perturbed with the curvature operator, precisely, the Laplace-Beltrami operator is replaced with "$…
Paper introduces a method for operator learning using random features.
problem Estimating maps between infinite-dimensional spaces using input-output pairs.
method Function-valued random features method, building a linear combination of random operators.
result The method provides convergence guarantees and error bounds for nonlinear problems.
This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept …
Proposes a new method to learn operators for stochastic problems using DeepONet with autoencoder.
problem Efficiently solve forward and inverse stochastic problems with limited data.
method MultiAuto-DeepONet, a multi-resolution autoencoder DeepONet model.
result The model effectively handles high-dimensional stochastic inputs and reduces the number of trainable parameters.
Fundamental solutions found for PDEs in Finsler geometry.
problem Solving nonlinear PDEs in Finsler geometry.
method Introduced a non-isotropic Minkowski gauge and computed fundamental solutions.
result Explicit fundamental solutions computed for the PDEs.
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.
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.
We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray transforms of lower order coefficients. We also give two different boundary determinat…
In this paper we present nonparametric estimators for coefficients in stochastic differential equation if the data are described by independent, identically distributed random variables. The problem is formulated as a nonlinear ill-posed operator equation with a deterministic forward operator described by the Fokker-Pl…
New method uses neural networks to solve high-dimensional eigenvalue problems.
problem Solving eigenvalue problems in high dimensions.
method Reformulates eigenvalue problem as fixed point problem of semigroup flow, approximated by neural networks.
result Accurate eigenvalue and eigenfunction approximations in various high-dimensional operators.
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.
Attention augments forest for tabular data accuracy.
problem Training tabular data models with high accuracy and efficiency.
method Tree Attention Block (TAB) in differentiable forest framework.
result Attention augmented differentiable forest achieves comparable and sometimes higher accuracy than GBDT models.
New machine learning methods solve complex PDEs with improved accuracy.
problem Solving fully nonlinear PDEs with convex Hamiltonian.
method Rewriting PDE in dual stochastic control form, estimating optimal feedback control with neural network, approximating value function with neural networks.
result Improved estimation of PDE solution and its derivatives, especially the second derivative.
Study on existence of metrics in conformal geometry with constraints on Schouten tensor.
problem Existence of metrics conformal to a given Riemannian metric with constraints on Schouten tensor.
method Use differential inclusions and viscosity solutions to prove existence and uniqueness of metrics.
result Existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor.
New method learns low-dimensional models for systems with non-polynomial terms.
problem Modeling systems with non-polynomial nonlinear terms that are spatially local and given in analytic form.
method Non-intrusive model reduction method that learns operators for linear and polynomially nonlinear dynamics via a least-squares problem incorporating given non-polynomial terms.
result Comparable accuracy to intrusive methods that require full knowledge of governing equations.
Infinitesimal variation of Action functional in classical (non-quantum) field theory with higher derivatives is presented in terms of well-defined intrinsic geometric objects independent of the particular field which varies. 'Integration by parts' procedure for this variation is then described in purely formal language…
This article provides a version of scale calculus geared towards a notion of (nonlinear) Fredholm maps between certain types of Frechet spaces, retaining as many as possible of the properties Fredholm maps between Banach spaces enjoy, and the existence of a constant rank theorem for such maps. It does so by extending t…
Constructs differential characters on nonlinear Graßmannians.
problem No specific problem stated; focuses on mathematical construction.
method Using a nonlinear version of the tautological bundle, a transgression map is constructed from M to nonlinear Graßmannians of submanifolds of fixed type. result Obtains prequantum circle bundles and central Lie group extensions.
The new business paradigms originate a strong necessity to re-think the theory of the firm with the aim to get a better understanding on the organizational and functional principles of the firm, operating in the investment economies in the prosperous societies. In this connection, we make the innovative research to adv…
We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where: (1) all we observe are …
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.
Scalable Gaussian Process Operator tackles high-dimensional PDEs.
problem Scaling Gaussian Process Operators to high-dimensional, data-intensive regimes.
method Nearest-neighbor-based local kernel approximations, sparse kernel approximation, structured Kronecker factorizations, operator-aware kernel structures, task-informed mean functions.
result Consistently achieves high accuracy across varying discretization scales.
We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.
Estimates for polynomial operators using determinant majorization and subharmonics.
problem Bounding solutions of polynomial operators on Euclidean domains.
method Combines Alexandrov estimate and determinant majorization, using subharmonics and semiconvex approximation.
result Includes classical Alexandrov-Bakelman-Pucci estimate for linear operators.
Physics-informed WNO learns PDE solutions without labeled data.
problem Data-hungry nature of WNO framework.
method Physics-informed WNO for learning PDE solutions.
result Validated and illustrated with four nonlinear systems.
Recently, the infinitesimal moduli space of heterotic G2 compactifications was described in supergravity and related to the cohomology of a target space differential. In this paper we identify the marginal deformations of the corresponding heterotic nonlinear sigma model with cohomology classes of a worldsheet BRST …
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.
Paper develops a novel approach for optimal control using kernel methods.
problem Optimal control of nonlinear stochastic systems.
method Infinitesimal generator approach in reproducing kernel Hilbert spaces.
result Data-driven solution to optimal control problems.
While there is currently a lot of enthusiasm about "big data", useful data is usually "small" and expensive to acquire. In this paper, we present a new paradigm of learning partial differential equations from {\em small} data. In particular, we introduce \emph{hidden physics models}, which are essentially data-efficien…