Paper discusses recent progress on fractional Laplacian in conformal geometry.
problem Fractional Laplacian in conformal geometry.
method Analytic and geometric approaches.
result Recent developments reported in both analytic and geometric perspectives.
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.
The paper finds a correspondence between invariant PDEs and hypersurfaces.
problem Constructing invariant PDEs on projective and affine spaces.
method Applying a general method to specific cases of projective and affine spaces.
result Projectively or affinely invariant PDEs correspond to CO(d, n-d) invariant hypersurfaces.
Paper designs Poisson integrators using machine learning.
problem Designing integrators that preserve Poisson geometry.
method Reformulated as an optimization problem in Hamilton-Jacobi PDE, solved using machine learning.
result Machine learning approximates solutions to Hamilton-Jacobi PDE.
Random feature model approximates PDE solutions efficiently.
problem Approximating solutions to PDEs with high-dimensional inputs and outputs.
method Random feature model applied to infinite-dimensional operators.
result Efficient and accurate approximation of PDE solutions.
A new method uses PDEs to predict spatiotemporal phenomena.
problem Predicting high-dimensional spatiotemporal data.
method Partial differential equations (PDEs) for spatiotemporal disentanglement.
result The method outperforms existing models in accuracy and applicability.
Bayesian PINN improves estimation of PDE solutions from noisy data.
problem Estimating solutions of PDEs from noisy measurements.
method Bayesian approach to Physics-informed neural networks (PINNs) for inverse problems.
result Convergence rate of Bayesian posterior mean error in PDE solutions.
This paper tackles collision avoidance for many UAVs using MFG and ML.
problem Collision avoidance for many UAVs in real-time missions.
method Mean-field game (MFG) theory combined with machine learning (ML) to reduce computation and communication energy.
result The proposed MFG learning control method achieves collision avoidance with low communication and acceptable computation energy.
Physics-informed GANs model subsurface flow at the Hanford Site.
problem Uncertainty quantification for subsurface flow modeling at the Hanford Site.
method Physics-informed GANs, hierarchical domain parallelism, multiple GPUs, efficient communication.
result Highly scalable physics-informed GANs model subsurface flow on Summit supercomputer.
New diagnostic method detects misspecified models in inverse PDE problems.
problem Misleading residual-norm diagnostics in inverse PDE problems.
method Structure-sensitive sequential diagnostic using e-processes.
result Rejects fitted models that produce biased predictions.
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.
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.
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 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.
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.
Solves second-order PDEs using quotients and differential invariants.
problem Solving second-order PDEs with first-order quotients.
method Solve the quotient PDE using differential invariants, then add new constraints to solve the original PDE.
result New method for solving second-order scalar PDEs with infinite-dimensional symmetry algebras.
PRISMA uses PDE residuals for fast, robust, and accurate inference.
problem Slow gradient-based optimization and instability in PDE residual-based methods.
method Integrates PDE residuals directly into the model's architecture via attention mechanisms in the spectral domain.
result Competitive accuracy with significantly lower inference costs and faster speeds.
Convert PDEs into Pfaffian fibrations for easier study.
problem Simplifying the study of PDEs.
method Encode PDE data into Pfaffian fibrations.
result Prolongations, integrability, and linearizations generalize to Pfaffian fibrations.
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.
Gamblets simplify solving complex implicit schemes for PDEs with rough coefficients.
problem Complexity bottleneck in solving implicit schemes for PDEs with rough coefficients.
method Generalized gamblets for near-linear complexity solution of implicit systems.
result Rigorous a-priori error bounds on numerical approximations of PDEs.
Paper uses deep learning to solve PDEs without supervision.
problem Solving elliptic PDEs without labeled data.
method Uses deep neural networks and least-squares functionals.
result Demonstrates effectiveness on 1D second-order elliptic PDEs.
Kernel method learns PDEs from noisy data.
problem Discovering and solving PDEs from noisy data.
method Kernel smoothing, regression, and operator learning.
result Competitive performance compared to state-of-the-art algorithms.
PDE-Net learns PDEs from data using neural networks.
problem Learning PDEs from complex system dynamics.
method Proposes PDE-Net, a feed-forward deep network to learn differential operators and nonlinear responses.
result PDE-Net can accurately predict dynamics and uncover hidden PDE models.
Solves a PDE for Landsberg surfaces using new Finsler surface insights.
problem Solving the Landsberg's PDE for Finsler surfaces.
method Reduces the system of non-linear PDEs to a single PDE, the Landsberg's PDE, and solves it.
result Obtains a class of solutions for the Landsberg's PDE.
Survey on conservation laws for geometric PDEs.
problem Modeling polyharmonic maps.
method Conservation law approach.
result Overview of conservation laws in geometric PDEs.
Using the theory of the symmetry group for PDEs [15, 17], we derive the symmetry group G associated to surfaces PDE. Several group invariant solutions of the surfaces PDE are given by solving a reduced system of partial differential equations.
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.
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.
Develops arithmetic PDE geometry concepts like curvature and cohomology.
problem Creating a geometry framework for arithmetic PDEs.
method Introducing arithmetic analogues of Levi-Civita and Chern connections, then developing curvature and characteristic classes.
result Arithmetic analogues of curvature and characteristic classes have been developed.
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.
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 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.
Method finds explicit solutions to certain PDEs.
problem Finding solutions to specific types of PDEs.
method Exploiting solvable structures to find explicit solutions.
result Effectiveness demonstrated on several examples.
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.
New method combines deep learning and splitting for high-dimensional PDEs.
problem Solving high-dimensional nonlinear parabolic PDEs efficiently.
method Combines operator splitting with deep learning for separate subproblems.
result Very good results in up to 10,000 dimensions with short run times.
We explore completely exceptional 2nd order scalar PDEs and their connection to Monge-Ampère equations.
problem Identifying a specific class of nonlinear PDEs that are not genuinely nonlinear.
method Unified geometric background review and definition of completely exceptional PDEs and Monge-Ampère equations.
result The class of completely exceptional 2nd order scalar PDEs reduces to Monge-Ampère equations.
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.
Automated PDE discovery from multiple noisy experiments.
problem Inherent variability in experiments makes single experiment inference unreliable.
method Randomised adaptive group Lasso sparsity estimator in deep learning framework.
result More generalizable PDEs found from multiple datasets.
VarNet solves PDEs with deep neural networks using variational loss.
problem Solving partial differential equations (PDEs) efficiently and accurately.
method VarNet uses a novel variational loss function and optimizes space-time samples for training deep neural networks.
result VarNet models are smooth, differentiable, and directly usable for PDE control and optimization.
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.
Meta-learning neural networks to solve diverse PDEs efficiently.
problem Efficiently solving new PDE problems with minimal training.
method Neural network meta-learning of PDE problem representations.
result Meta-learned neural networks predict PDE solutions with high accuracy.
Discover PDEs from complex data using machine learning.
problem Discovering PDEs from real-world data without first principles.
method Machine learning, deep learning, reverse engineering PDEs from data.
result Deep learning can accurately discover PDEs from complex data.
Nonlinear PDEs in finance optimization problems solved numerically and analytically.
problem Nonlinear PDEs in finance optimization problems.
method Numerical and analytical solutions.
result Solutions to specific PDEs in finance.
New method solves high-dimensional Kolmogorov PDEs without curse of dimensionality.
problem Solving high-dimensional Kolmogorov PDEs efficiently and accurately.
method Deep learning-based numerical approximation method.
result Effective numerical approximation of Kolmogorov PDEs in high dimensions.
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 …
The paper extends geometric study to systems of PDEs, introducing variational bi-complex for conservation laws.
problem Geometric study of systems of semi-linear hyperbolic PDEs in three variables.
method Introduces variational bi-complex to define form-valued conservation laws and provides a method for generating conservation laws.
result Generates infinitely many conservation laws for systems of three equations.
The paper studies geometric PDEs for flatness on Riemannian manifolds.
problem Understanding flatness in geometric PDEs.
method Study geometric PDEs of connection-flatness, curvature-flatness, Ricci-flatness, scalar curvature-flatness.
result Introduce new Theorems about flatness in Differential Geometry.
Proposes MscaleDNN for solving high-dimensional PDEs efficiently.
problem Solving high-dimensional PDEs efficiently.
method Radial scaling in frequency domain and compact support activation functions.
result Increased power in multi-scale resolution and high frequency capturing.