Constructs a moduli space for PDEs, linking stability to geometric metrics.
problem Moduli space construction for involutive ideal sheaves from PDEs.
method Introduces D \mathcal{D} D -Hilbert and D \mathcal{D} D -Quot functors, defines Spencer stability. result Spencer poly-stability of PDE ideal implies Hermitian-Yang-Mills metric existence.
Paper solves PDEs for optimal investment strategies in volatile markets.
problem Finding optimal investment strategies in volatile markets.
method Numerical methods using time-changed Bessel bridges.
result Solves PDEs for relative arbitrage opportunities in volatility-stabilized markets.
Novel approach ensures stability of compact schemes for variable PDEs.
problem Ensuring stability of compact schemes for variable coefficient PDEs.
method Difference equation approach to derive stability conditions.
result Derives sufficient condition for unconditional stability.
Study on stability of geodesic maps in non-isotropic manifolds.
problem Stability of totally geodesic wave maps in non-isotropic manifolds.
method Factorization property, PDE system in geodesic normal coordinates, global existence result via hyperboloidal foliation.
result Established global existence for small initial data, leading to geometric stability.
Proves stability of convex disks close to round caps.
problem Stability of convex disks with positive curvature and strictly convex boundary.
method Compactness result for a Liouville-type PDE problem.
result Proves stability for a theorem of F. Hang and X. Wang.
The stability and robustness of compact schemes for parabolic PDEs are analyzed.
problem Stability and robustness of compact schemes for solving parabolic PDEs.
method Compact spatial discretization, Crank-Nicolson temporal discretization, eigenvalue analysis of amplification matrix.
result An upper bound on the condition number of the amplification matrix is derived, showing stability.
Local Neural Operators enable efficient system-level analysis of complex PDEs.
problem System-level analysis of large-scale dynamical systems using neural operators.
method Integrating local Neural Operators with Krylov subspace iterative methods for stability and bifurcation analysis.
result Demonstrated effectiveness of local Neural Operators in fixed-point, stability, and bifurcation analysis of nonlinear PDEs.
A method constructs invariant PDEs on homogeneous manifolds.
problem Finding invariant PDEs on homogeneous manifolds.
method Describes a general method for constructing invariant PDEs by reducing the problem to invariant hypersurfaces under the action of the stability subgroup.
result Describes invariant PDEs for hypersurfaces in Euclidean and conformal spaces.
This paper deals with stability in the numerical solution of the prominent Heston partial differential equation from mathematical finance. We study the well-known central second-order finite difference discretization, which leads to large semi-discrete systems with non-normal matrices A. By employing the logarithmic sp…
GRAND treats GNNs as PDE discretizations, addressing graph learning issues.
problem Graph learning issues like depth, oversmoothing, and bottlenecks.
method Models GNNs as a continuous diffusion process, treating them as PDE discretizations.
result Linear and nonlinear versions of GRAND achieve competitive results on graph benchmarks.
Let M M M be a manifold, V V V be a vector field on M M M , and B B B be a Banach space. For any fixed function f : M → B f:M\rightarrow B f : M → B and any fixed complex number λ λ λ , we study Hyers-Ulam stability of the global differential equation V y = λ y + f Vy=λy+f V y = λ y + f .
Stability results for geometric equations in warped product spaces.
problem Geometric partial differential equations in warped product spaces.
method Stability theorem development for level sets of functions.
result Quantitative stability theorems for Serrin's problem and Alexandroff's theorem.
Error estimates for nonlinear PDEs using kernel/GP methods.
problem Error analysis of kernel/GP methods for nonlinear and parametric PDEs.
method Sobolev space error estimates based on minimizing norm property of the solution.
result Dimension-benign convergence rates for smooth solutions.
Applications in quantitative finance such as optimal trade execution, risk management of options, and optimal asset allocation involve the solution of high dimensional and nonlinear Partial Differential Equations (PDEs). The connection between PDEs and systems of Forward-Backward Stochastic Differential Equations (FBSD…
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.
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.
A lightweight framework improves convergence and stability of PINNs for complex PDEs.
problem Training instability and reduced accuracy in PINNs for complex PDEs.
method Adaptive curvature correction using secant information to optimize first-order optimizers.
result Consistent improvements in convergence speed, stability, and accuracy over standard optimizers.
Survey on recent developments in isometric immersions using PDE techniques.
problem Analyzing isometric immersions with low Sobolev regularity.
method Compensated compactness and Coulomb-Uhlenbeck gauges.
result Weak continuity and stability of Gauss-Codazzi-Ricci equations.
Letter analyzes training dynamics of a nonlinear contrastive learning model in high dimensions.
problem Understanding training dynamics of nonlinear contrastive learning models in high-dimensional settings.
method High-dimensional analysis using McKean-Vlasov PDEs and low-dimensional ODEs.
result The model's performance evolves according to specific ODEs, revealing features like feature learnability and noise effects.
Unified estimate for complex Monge-Ampère equations on Kähler manifolds.
problem Estimating solutions to complex Monge-Ampère equations on Kähler manifolds.
method Unified approach using PDE methods and entropy bounds to construct comparison metrics.
result Improves previous results on modulus of continuity, stability, and W 1 , 1 W^{1,1} W 1 , 1 -estimates of Green's functions. This work introduces a new loss function to improve the efficiency of optimization-based PDE solvers.
problem Optimization-based PDE solvers converge slowly and are inefficient compared to classical iterative solvers.
method Proposes a novel Stabilized Gradient Residual (SGR) loss function to modulate the condition number.
result The SGR loss achieves orders-of-magnitude faster convergence than the MSE loss in both ODIL and PINNs frameworks.
Geometric approach to moment maps in complex geometry.
problem Constructing moment maps in complex geometry.
method Introducing universal families and equivariant differential forms.
result New geometric proofs and equations for moment maps.
Study stabilizes translating solitons in hyperbolic space for MCF.
problem Stability of translating solitons in hyperbolic space.
method Developed theory, constructed rotationally invariant translators, used avoidance principle and maximum principle.
result Horospheres are dynamically stable as radial graphical solutions to MCF.
New framework discovers PDEs from sparse, noisy data.
problem Discovering PDEs with high-order derivatives and heterogeneous parameters.
method Combines deep-learning and integral form to handle sparse and noisy data.
result More robust and accurate compared to existing methods.
Develops a mean-field theory for multi-head self-attention under cross-entropy training.
problem Mean-field analysis of multi-head self-attention under cross-entropy training.
method Mean-field theory for a simplified single-layer causal multi-head self-attention model.
result Proves a static finite-head approximation bound for the optimal risk.
Adapts Stein's method for geometric inequalities, addressing boundary terms.
problem Geometric inequalities and their stability under constraints.
method Uses elliptic PDE with oblique boundary condition to handle boundary terms.
result Stability results for various geometric inequalities with respect to a new distance.
Enhances neural network solvers for PDEs with complex boundary conditions.
problem Challenges in solving PDEs with high accuracy and complex boundary conditions.
method Integrates natural gradient optimization with numerical time-stepping schemes to enforce Dirichlet boundary conditions.
result Superior accuracy and computational efficiency of the proposed methods for solving PDEs.
Study geometric flow on curves with positive torsion, finding stationary solutions and their stability.
problem Analyzing geometric flow on curves with positive torsion.
method Evolution equation X t = 1 τ e x t b f B X_{t}=\frac{1}{\sqrtτ} extbf{B} X t = τ 1 e x t b f B , studying stationary solutions and linear stability. result Explicit formula for stationary solutions of helices with constant curvature and torsion, proving stability.
New algorithm solves Schrödinger bridge problem with mismatched channels.
problem Solving Schrödinger bridge problem with input and noise channel mismatch.
method Design of a Sinkhorn recursion with memory for nonlinear PDEs.
result Demonstrates solving control-affine Schrödinger bridge problem.
Proposes a method to refine PDE-driven high-dimensional rare-event simulation.
problem Challenges in constructing accurate surrogates for rare-event simulation.
method Adaptive importance sampling framework that refines a locally constructed surrogate.
result Achieves accuracy comparable to true-model adaptive importance sampling with fewer high-fidelity evaluations.
Physics-informed GANs estimate elastic moduli from mechanical tests.
problem Estimating spatially-varying elastic moduli from measured deformations.
method Physics-informed Generative Adversarial Networks (PI-GANs) with PDE constraints.
result Generated stiffness samples match true distribution statistics.
CPCMs integrate causal drivers for robust portfolio optimization.
problem Degradation of classical portfolio models under structural breaks and lack of arbitrage consistency in machine learning.
method Causal PDE-Control Models integrating structural causal drivers, nonlinear filtering, and forward-backward PDE control.
result CPCM solvers achieve higher Sharpe ratios and lower turnover than benchmarks.
We provide an introduction to the mathematics and physics of the deformed Hermitian-Yang-Mills equation, a fully nonlinear geometric PDE on Kahler manifolds which plays an important role in mirror symmetry. We discuss the physical origin of the equation, and some recent progress towards its solution. In dimension 3 we …
The paper solves a thermodynamics problem about crystal shape.
problem Understanding if minimizing free energy with convex potential and mass constraint generates a convex crystal.
method Utilized a stability theorem, convexity, and a new maximum principle approach to prove a three-dimensional convexity theorem.
result Completely settled the Almgren problem in R 3 \mathbb R^3 R 3 under generic conditions. DeepSVM learns SVMs without PDE solving, achieving high pricing accuracy.
problem Computational bottleneck in real-time calibration of stochastic volatility models.
method Physics-informed Deep Operator Network (PI-DeepONet) that enforces terminal payoffs and no-arbitrage conditions.
result DeepSVM achieves high pricing accuracy across various market dynamics.
New method stabilizes machine learning for physics-informed inverse problems.
problem Reconstructing physical quantities from PDE-compliant measurements.
method Physics-informed learning with smooth inductive bias.
result PDE operators stabilize variance and prevent overfitting in fixed dimensions.
Improved spectral convergence bounds for diffusion maps on tori.
problem Weak theoretical error bounds for diffusion maps.
method Spatial Hardy space estimates, PDE spectral stability, Sinkhorn weights.
result Matched pointwise error bounds for spectral data and operator convergence.
We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions n + 1 ≥ 3 n+1\ge 3 n + 1 ≥ 3 , and all have a point singularity where the curvature blows up; their evolution under the Ricci flow is in sh…
Robust PDE method for path-dependent Asian-style options using MPDATA.
problem Valuation of path-dependent Asian-style options.
method Non-oscillatory forward-in-time second-order MPDATA finite-difference scheme for solving 2D PDEs.
result MPDATA scheme improves solution over first-order upwind step, highlighting its importance.
Center manifold analysis can be used in order to investigate the stability of the stationary solutions of various PDEs. This can be done by considering the PDE as an ODE between certain Banach spaces and linearising about the stationary solution. Here we investigate the volume preserving mean curvature flow using such …
DCGD improves training of PINNs by adjusting gradients to avoid negative inner products.
problem Pathological behaviors in PINNs training, especially gradient imbalance.
method Dual Cone Gradient Descent (DCGD) framework to adjust gradient direction.
result DCGD outperforms other optimization algorithms in various evaluation metrics.
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.
Characterizes solutions to Z-critical equations on surfaces using effective conditions.
problem Characterizing solutions to Z-critical equations on compact Kähler surfaces.
method Uses effective conditions and Picard number bounds to characterize solutions.
result Characterizes optimally destabilizing curves for Donaldson's J-equation and deformed Hermitian Yang-Mills equation.
New method learns PDE solutions from low-fidelity data.
problem Challenges in learning PDE surrogates with scarce data.
method Flow matching in infinite-dimensional space with conditional neural operators.
result Accurately learns PDE solutions across different resolutions and fidelities.
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.
This paper explores the vol-of-vol parameter in the Heston model and its relation to VVIX.
problem Calibrating the Heston model to market data for stable exotic option pricing.
method Four approaches to estimate VVIX in the Heston model: transition density, analytical approximation, and PDE-based.
result Improved calibration stability of the Heston model using the estimated VVIX.
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.
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 …