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.
Extends Fredholm theory to Frechet spaces.
problem Extending Fredholm theory to Frechet spaces.
method Extending linear Fredholm maps to Frechet spaces using Nash-Moser inverse function theorem.
result Existence of a constant rank theorem for nonlinear Fredholm maps in Frechet spaces.
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. 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<∞. 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.
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 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.
We establish a microscopic convexity principle for nonlinear elliptic and parabolic partial differential equations in general form.
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.
Some of recent developments, including recent results, ideas, techniques, and approaches, in the study of degenerate partial differential equations are surveyed and analyzed. Several examples of nonlinear degenerate, even mixed, partial differential equations, are presented, which arise naturally in some longstanding, …
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.
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.
New method learns physics equations from small data.
problem Learning partial differential equations from limited data.
method Hidden physics models using Gaussian processes.
result Data-efficient learning of complex equations.
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 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.
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.
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.
New PDEs of mixed type emerge in fluid mechanics and geometry.
problem Analysis of nonlinear PDEs of mixed type.
method Through historical problems and recent trends.
result Many PDEs are of mixed type, requiring new analysis.
Online algorithm identifies PDEs from noisy data snapshots.
problem Identifying PDEs from sequential solution snapshots.
method Combines weak-form discretization with online proximal gradient descent.
result Efficiently identifies and tracks systems with time-varying coefficients.
A new method for solving time-dependent PDEs using Gaussian processes.
problem Solving time-dependent and non-linear PDEs with noisy data and uncertainty quantification.
method Numerical Gaussian processes with covariance functions from time-dependent PDEs, using Gaussian process priors for temporal discretization.
result Accurate approximations of latent solutions with uncertainty propagation, even for long time integration.
Study on nonlinear equations on spheres and hemispheres with zero Neumann boundary condition.
problem Finding conditions for constant solutions to Brezis-Nirenberg type problems.
method Developed a study involving nonlinear partial differential equations on spheres and hemispheres with zero Neumann boundary condition.
result Conditions for equations to have only constant solutions.
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.
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.
Physics-informed neural networks solve physics problems using neural nets.
problem Discovering nonlinear PDEs from data.
method Two classes of algorithms: continuous time and discrete time models.
result Demonstrated effectiveness on various physics problems.
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.
Generalizes neural networks for infinite-dimensional mappings, including PDE solutions.
problem Learning mappings between infinite-dimensional spaces and finite-dimensional approximations.
method Graph kernel network architecture with message passing for kernel integration.
result Competitive performance compared to state-of-the-art solvers for PDEs.
Determines algebra structure of complex differential forms operators.
problem Identifying the algebra structure of differential operators on complex-valued differential forms.
method Shows it is the universal enveloping algebra of a graded Lie algebra and determines its cohomology.
result Determines the cohomology of the graded Lie algebra with respect to various inner differentials.
This paper surveys some recent results on existence, uniqueness and removable singularities for fully nonlinear differential equations on manifolds. The discussion also treats restriction theorems and the strong Bellman principle.
Paper surveys Kronecker webs and solves bisymplectic realization problem.
problem Bisymplectic realization of bihamiltonian structures.
method Partial Nijenhuis operator approach.
result Partial solution to bisymplectic realization problem.
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.
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.
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.
Rediscovered by a systematic search, a forgotten class of integrable surfaces is shown to disprove the Finkel-Wu conjecture. The associated integrable nonlinear partial differential equation zyy+(1/z)xx+2=0 possesses a zero curvature representation, a third-order symmetry, and a nonlocal transformatio…
Physics-informed neural networks solve PDEs using neural networks.
problem Solving nonlinear partial differential equations (PDEs) with neural networks.
method Physics-informed neural networks trained to solve PDEs while respecting physical laws.
result Physics-informed neural networks can infer solutions to PDEs and create differentiable surrogate models.
Paper introduces FNM framework for learning finite-dimensional parametrized models.
problem Efficiently learning finite-dimensional parametrized models from limited data.
method Fourier Neural Mappings (FNMs) framework for operator learning.
result End-to-end learning of PtO maps can be less data-efficient than learning the solution operator first.
Novel approach classifies conformally equivariant spinor operators.
problem Classifying conformally equivariant differential operators on spinors.
method Classification based on vector-valued partial differential equations and spin Howe duality.
result Solutions for a vector-valued system of PDEs associated with D-modules. New theorems prove uniqueness of solutions to geometric PDEs.
problem Proving uniqueness of solutions to geometric PDEs.
method Analyzing nonlinear elliptic PDEs of divergence form.
result Proved several Moser-Bernstein type theorems.
New algorithm solves high-dimensional PDEs and BSDEs using neural networks.
problem Solving high-dimensional PDEs and BSDEs efficiently and accurately.
method Analogy with reinforcement learning, neural network approximation of policy function.
result Efficiency and accuracy demonstrated in solving 100-dimensional equations.
Physics-informed DeepONets solve PDEs without paired data, predicting solutions quickly.
problem Lack of paired input-output data for solving PDEs.
method Physics-informed DeepONets use automatic differentiation to enforce physical laws as soft penalty constraints.
result Physics-informed DeepONets can solve PDEs without paired data, predicting solutions up to 3 orders of magnitude faster.
A new method uses multifidelity Gaussian process regression to solve nonlinear PDEs.
problem Efficiently solving nonlinear PDEs using kernel methods.
method Proposes a kernel learning approach based on cokriging for multifidelity simulations.
result Demonstrates improved performance on the Burgers' equation.
Paper solves complex control problems using novel SDEs.
problem Solving stochastic differential games for nonlinear systems.
method Uses Deep Forward-Backward SDEs with neural networks.
result Numerical solution validated on two example systems.
Uniform bounds for complex equations using Monge-Ampère method.
problem Bounding solutions to complex equations.
method Auxiliary Monge-Ampère equation method.
result Uniform bounds remain valid even as background metrics degenerate.
FNOs improve spatio-temporal forecasting without needing PDE details.
problem Complex spatio-temporal dynamics in physical and biological phenomena.
method Fourier Neural Operators (FNOs) for dynamic spatio-temporal modeling.
result FNO forecasts are accurate and capture complex real-world dependencies.
PDE-Net 2.0 learns PDEs from data without prior knowledge.
problem Discovering PDEs from empirical data without detailed prior knowledge.
method Numeric-symbolic hybrid deep network combining numerical approximations and symbolic neural networks.
result PDE-Net 2.0 can uncover hidden PDEs and predict dynamics in noisy environments.
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.
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
The paper uses symmetry groups to simplify observability analysis of PDEs.
problem Observability of nonlinear PDEs with input and output.
method Differential-geometric representation and symmetry groups to transform solutions.
result Conditions for existence of symmetry groups that preserve input and output trajectories.
Deep learning discovers physics laws from data.
problem Discovering nonlinear PDEs from noisy data.
method Two deep neural networks: prior and dynamics.
result Accurately learns dynamics and forecasts future states.