Meta-learning base distributions for efficient PDE solutions.
problem Efficiently solving parametric parabolic PDEs across different scenarios.
method Meta-learning base distributions to compute PDE solutions.
result Improves generalization to new parameter regimes.
It is shown that the characteristic vector field associated to a first order PDE has the same form of an infinitesimal generator of an odd-symplectic transformation with contact Hamiltonian the given PDE. It is considered under which condition such PDE has a characteristic vector field commuting with a generator of an …
Generative model improved using Liouville PDE-based sliced-Wasserstein flow.
problem Improving generative models for fair regression.
method Transformed sliced-Wasserstein flow into Liouville PDE-based formalism, handling density estimation with normalizing flows of neural ODE.
result Outperforms in convergence and fairness with reduced variance.
Unified framework solves nonlinear PDEs and IPs using Gaussian processes.
problem Solving and identifying parameters in nonlinear PDEs and inverse problems.
method Gaussian process framework approximating solutions as MAP estimators, reducing to finite-dimensional optimization problem.
result Unified method converges in a small number of iterations for various PDEs.
The paper proved that every C2-solution of a given first order PDEs system, regarded on the jet fibre bundle of order one J1(T,M), may be viewed as a "generalized harmonic map", via the least squares variational method. Our ideas are structured in the following way: 1) we find a suitable geometrical structure on …
Neural Q-learning tackles high-dimensional PDEs.
problem Solving high-dimensional PDEs is computationally challenging.
method Adapting Q-learning from reinforcement learning to solve PDEs.
result The neural network approximator converges to the PDE solution as the network width increases.
Framework estimates PDEs from noisy data using neural networks.
problem Estimating unknown PDEs from noisy data.
method Interpolates noisy samples using a neural network, extracts PDE by matching derivatives.
result Method outperforms other methods in low signal-to-noise regimes.
Improved neural PDEs trained on augmented data enhance model accuracy and efficiency.
problem Training neural PDEs on limited data to accurately represent complex systems.
method Space-filling sampling of local states to generate augmented training data.
result Data-augmented neural PDEs outperform traditional emulators in accuracy and stability.
PDE-NetGen converts physical equations to neural networks for various scientific problems.
problem Bridging physics and deep learning for efficient neural network architectures.
method Combines symbolic calculus and neural network generation to translate PDEs into NN architectures.
result Generates compact, computationally-efficient physics-informed NN architectures.
PDE-based G-CNNs add geometric symmetries to CNNs without augmentation.
problem Designing CNNs with built-in symmetries like rotation.
method Formulate CNN layers as PDE solvers on homogeneous spaces.
result PDE-G-CNNs achieve better performance with fewer parameters.
The paper presents a PDE method for xVA incorporation in financial derivatives.
problem Incorporating value adjustments (xVA) in financial derivative pricing.
method Analytical solution of PDEs in the Black-Scholes framework.
result New semi-closed formulas for xVA are derived and compared to Monte-Carlo and numerical methods.
New integrable matrix PDEs derived from Frölicher-Nijenhuis brackets.
problem Developing integrable systems from tensor field properties.
method Using Frölicher-Nijenhuis brackets to generate bi-differential graded algebras and PDE systems.
result New integrable nonlinear matrix PDEs and systems are derived.
Implicit schemes are popular methods for the integration of time dependent PDEs such as hyperbolic and parabolic PDEs. However the necessity to solve corresponding linear systems at each time step constitutes a complexity bottleneck in their application to PDEs with rough coefficients. We present a generalization of ga…
The aim of this paper is fourfold. Firstly, we introduce and study the f-ultra-harmonic maps. Secondly, we recall the geometric dynamics generated by a first order normal PDE system and we give original results regarding the geometric dynamics generated by other first order PDE systems. Thirdly, we determine the Gauss …
The paper studies third-order PDEs invariant under affine transformations and connects them to the Fubini-Pick invariant.
problem Investigating third-order PDEs invariant under affine transformations.
method Using a general method introduced in [D.V. Alekseevsky, J. Gutt, G. Manno, and G. Moreno: A general method to construct invariant PDEs on homogeneous manifolds].
result Derives third-order PDEs from the Fubini-Pick invariant.
Derives PDEs for pricing RFR derivatives under a new FMM model.
problem Valuation of interest rate derivatives under a new FMM model.
method Develops PDEs and finite differences methods for numerical solution.
result First use of PDE methods for RFR derivatives valuation.
Develops numerical methods for PDEs on hypergraphs and networks.
problem Solving PDEs on complex geometric structures like hypergraphs and networks.
method Hybrid finite element methods, focusing on hybrid discontinuous Galerkin methods.
result Derives numerical approximations for PDEs on hypergraphs and networks.
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
DINo forecasts PDEs with flexible extrapolation and adaptability.
problem Fixed discretizations limit real-world PDE forecasting.
method DINo uses implicit neural representations for continuous-time dynamics.
result DINo outperforms other neural PDE forecasters.
DL-PDE discovers PDEs from noisy, sparse data using neural networks and sparse regressions.
problem Discovering PDEs from noisy, sparse data.
method Combines neural networks and sparse regressions to discover PDEs from meta-data generated by a neural network.
result Achieves satisfactory results in real-world engineering settings with noisy and limited data.
New method reduces PDE surrogate model training costs by selectively acquiring time steps.
problem High computational cost of generating training data for PDE surrogate models.
method STAP (Selective Time-Step Acquisition for PDEs) framework that acquires only important time steps.
result Demonstrated effectiveness on several benchmark PDEs, reducing training costs.
PANIS learns PDE surrogates for heterogeneous materials without solving the PDE.
problem Learning surrogates for parametrized PDEs in heterogeneous media.
method Physics-aware neural implicit solvers combining probabilistic learning and physics-informed discretization.
result Learned surrogates for effective solutions in heterogeneous materials without solving the reference problem.
INFERS PDEs from data samples using learned context.
problem Inferring explicit PDEs from unseen dynamics.
method Contextual Finite Differences (CFD) method integrating PDE form and differential scheme.
result Yields a PDE fitting the data sample for signal prediction and explanation.
Paper analyzes DRM for solving high-dimensional elliptic PDEs with generalization bounds.
problem Analyzing generalization error of neural network methods for high-dimensional PDEs.
method Developed a new solution theory for spectral Barron space and derived generalization error bounds.
result Generalization error bounds are independent of dimension and solutions lie in spectral Barron space.
FM4PDE learns PDE solutions from sparse data.
problem Reconstructing PDE solutions from limited observations.
method Flow-matching generative framework that learns PDE coefficients and solutions.
result Error guarantees for guided procedures, including deterministic and stochastic samplers.
A new method infers parameters from PDEs using Gaussian processes.
problem Estimating unknown parameters in PDEs from noisy data.
method PDE-Informed Gaussian Process (PIGP) method.
result The method bypasses numerical solvers for PDEs and provides uncertainty quantification.
GIT-Net uses neural networks to approximate PDE operators efficiently.
problem Approximating PDE operators for complex geometries.
method Parametrizes adaptive generalized integral transforms with deep neural networks.
result GIT-Net outperforms existing neural network operators in multiple areas.
New method solves high-dimensional PDEs fast using physics-informed neural networks.
problem High computational cost in solving high-dimensional PDEs.
method Stochastic Dimension Gradient Descent (SDGD) for physics-informed neural networks (PINNs).
result Solves many high-dimensional PDEs including HJB and Schrödinger equations in 100,000 dimensions in 12 hours.
New method uses Gaussian processes to improve PDE solver accuracy.
problem Uncertainty in PDE solver parameters and measurements.
method Physics-informed Gaussian process regression.
result Strictly generalizes weighted residual methods.
New framework uses PDE for no-regret generative modeling.
problem Developing efficient generative models for complex distributions.
method Iterative refinement of Brenier maps using mirror gradient descent.
result Converges to optimal Brenier map under various step-size schedules.
Neural-PDE learns PDEs from data using LSTM, outperforming traditional methods.
problem Solving time-dependent PDEs numerically is challenging.
method Bidirectional LSTM encoder to learn governing rules from data.
result Neural-PDE efficiently predicts PDE dynamics with minimal parameters.
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.
Data-efficient PDE operator learning without expensive simulations.
problem Expensive numerical PDE solutions limit data efficiency in machine learning.
method Unsupervised pretraining and in-context learning.
result Highly data-efficient and more generalizable than conventional models.
Solves natural PDEs for minimal Lorentz surfaces in 4D spacetime.
problem Natural PDEs for minimal Lorentz surfaces in R24. method Weierstrass type representations and canonical coordinates.
result Explicit solution of the system of natural PDEs.
Physics-informed neural networks and neural operators speed up solving parametric PDEs by orders of magnitude.
problem Solving PDEs for varying parameters is computationally expensive.
method Physics-informed neural networks and neural operators learn solution mappings across parameter spaces.
result Neural operators achieve computational speedups of 10^3 to 10^5 times faster than traditional methods.
Enhances uncertainty modeling in random PDEs using PINNs and generative models.
problem Uncertainty in complex systems modeled by random PDEs.
method Combines Physics-Informed Neural Networks (PINNs) with generative modeling techniques.
result Systematic control of uncertainty with maintained predictive accuracy.
Partial differential equations (PDEs) are commonly derived based on empirical observations. However, recent advances of technology enable us to collect and store massive amount of data, which offers new opportunities for data-driven discovery of PDEs. In this paper, we propose a new deep neural network, called PDE-Net …
The paper explores symmetry in solutions of semilinear PDEs on Riemannian domains.
problem Symmetry phenomena in solutions of semilinear PDEs on Riemannian domains.
method General framework for formulating the symmetry problem; evidence from stable solutions; consideration of manifolds with density.
result Evidence that the framework is natural, with results for stable solutions.
Efficiently optimizes hyperparameters for PDE and inverse problems using Gaussian processes.
problem Hyperparameter optimization for scientific computing and inference methods.
method Bilevel optimization with Gauss-Newton linearization for efficient hyperparameter updates.
result Significant improvements in accuracy and robustness compared to random initialization.
Enhances neural operators with physics knowledge for more accurate simulations.
problem Improving accuracy and generalization of neural operators for physical systems.
method Jointly learns from original PDEs and simplified forms, incorporating fundamental physics.
result Significant improvement in nRMSE across various PDE problems.
We proved that the solutions of C2 class of certain ODEs or PDEs belong to a class of harmonic maps between two convenient generalized Lagrange spaces.
New PDE systems generalize Hawking mass monotonicity.
problem Generalizing Hawking mass monotonicity to initial data sets.
method Introduced new systems of PDE on initial data sets (M,g,k). result Generalized Geroch's monotonicity formula to initial data sets.
This paper extends, to a class of systems of semi-linear hyperbolic second order PDEs in three variables, the geometric study of a single nonlinear hyperbolic PDE in the plane as presented in [Anderson I.M., Kamran N., Duke Math. J. 87 (1997), 265-319]. The constrained variational bi-complex is introduced and used to d…
A framework for reducing PDEs by symmetry, preserving key structures.
problem Reducing PDEs while preserving geometric structures and symmetries.
method Systematic calculation of reduced forms for various geometric structures.
result Noether's theorem is inherited in reduced systems, preserving conservation laws.
Proposes ENOs for learning PDE solutions that conserve energy.
problem Learning dynamics that obey physical laws, especially in super-resolution settings.
method Energy-consistent Neural Operators (ENOs) with a novel penalty function inspired by energy-based theory.
result ENOs outperform existing DNN models in predicting solutions from data, especially in super-resolution settings.
PDE-DKL combines NNs and GPs for high-dimensional PDE problems.
problem High-dimensional PDE problems with scarce data.
method PDE-constrained Deep Kernel Learning (PDE-DKL) framework.
result High accuracy with reduced data requirements.
Deep learning for HJB PDEs using synthetic data and residual minimization.
problem Solving Hamilton-Jacobi-Bellman PDEs for optimal control problems.
method Gradient-augmented synthetic dataset for supervised learning, residual minimization.
result Improves accuracy and efficiency of deep learning for HJB PDEs.