Study rough volatility models using path-dependent PDEs and fractional Brownian motions.
problem Modeling and analyzing rough volatility in financial markets.
method Showed conditional expectations are unique classical solutions to path-dependent PDEs derived from functional Itô formula. Leverage these to study weak rates of convergence for discretized stochastic integrals.
result Obtained optimal weak error rates for approximating log-stock prices in rough volatility models.
We prove that the natural principal parameters on a given Weingarten surface are also natural principal parameters for the parallel surfaces of the given one. As a consequence of this result we obtain that the natural PDE of any Weingarten surface is the natural PDE of its parallel surfaces. We show that the linear fra…
New method learns fractional order of PDEs from flocking particle simulations.
problem Deriving effective nonlocal influence functions from discrete agent-based models.
method Agent-based model, fractional PDEs, Gaussian process regression, Bayesian optimization.
result Learned Euler equations accurately predict flocking behavior.
The aim of this paper is to report on recent development on the conformal fractional Laplacian, both from the analytic and geometric points of view, but especially towards the PDE community.
The fractional Yamabe problem, proposed by González-Qing (2013, Anal. PDE) is a geometric question which concerns the existence of metrics with constant fractional scalar curvature. It extends the phenomena which were discovered in the classical Yamabe problem and the boundary Yamabe problem to the realm of nonlocal co…
Paper approximates fractional harmonic maps with numerical methods.
problem Approximating fractional harmonic maps with constraints and nonlocality.
method Weak compactness results and numerical methods for various PDEs.
result Convergence of numerical approximations for fractional harmonic maps.
Paper proves short-term existence of fractional mean curvature flow.
problem Existence of solutions for fractional mean curvature flow with capillary boundary conditions.
method Fixed point argument.
result Short time existence of solutions for fractional mean curvature flow.
Data-driven discovery of "hidden physics" -- i.e., machine learning of differential equation models underlying observed data -- has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a…
Let X be an asymptotically hyperbolic manifold and M its conformal infinity. This paper is devoted to deduce several existence results of the fractional Yamabe problem on M under various geometric assumptions on X and M: Firstly, we handle when the boundary M has a point at which the mean curvature is negat…
Unified framework for forward and inverse PDE problems in multiphase media.
problem Non-differentiable inverse problems in discrete-valued material fields.
method GenPANIS: Latent-variable generative framework preserving discrete microstructures.
result Unified bidirectional inference with minimal labeled pairs and physics-aware decoder.
On any space-like W-surface in the three-dimensional Minkowski space we introduce locally natural principal parameters and prove that such a surface is determined uniquely up to motion by a special invariant function, which satisfies a natural non-linear partial differential equation. This result can be interpreted as …
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.
DeepXDE uses neural networks to solve complex differential equations.
problem Solving differential equations using machine learning.
method Physics-informed neural networks (PINNs) with adaptive refinement.
result PINNs can solve various types of PDEs and inverse problems.
We study Sobolev-type metrics of fractional order s≥0 on the group $\Diff_c(M)$ of compactly supported diffeomorphisms of a manifold M. We show that for the important special case M=S1 the geodesic distance on $\Diff_c(S^1)$ vanishes if and only if s≤21. For other manifolds we obtain a partial chara…
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.
We show by explicit closed form calculations that a Hurst exponent H that is not 1/2 does not necessarily imply long time correlations like those found in fractional Brownian motion. We construct a large set of scaling solutions of Fokker-Planck partial differential equations where H is not 1/2. Thus Markov processes, …
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.
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.
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.
In this work we present a new approach on studying dynamical systems. Combining the two ways of expressing the uncertainty, using probabilistic theory and credibility theory, we have research the generalized fractional hybrid equations. We have introduced the concepts of generalized fractional Wiener process, generaliz…
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.
Methods from the geometry of nonholonomic manifolds and Lagrange-Finsler spaces are applied in fractional calculus with Caputo derivatives and for elaborating models of fractional gravity and fractional Lagrange mechanics. The geometric data for such models are encoded into (fractional) bi-Hamiltonian structures and as…
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.
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.
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.
Introduces fractional k-dimensional measure bridging fractional length and area.
problem Defining fractional measures for dimensions between 0 and n-1.
method Introduces a parameterized fractional measure σ that converges to Hausdorff measure. result Fractional measure converges to Hausdorff measure with a known constant factor.
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.
The theory of derivative of noninteger order goes back to Leibniz, Liouville and Riemann. Derivatives of fractional order have found many applications in recent studies in mechanics, physics, economics. In this paper we define the fractional tangent bundle on a manifold, using a method of Radu Miron. The fractional Lei…
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 …