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.
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.
Space mapping speeds up shape optimization for PDEs.
problem Efficiently solving shape optimization problems constrained by PDEs.
method Combines fine and coarse model optimizations using Riemannian metrics.
result Space mapping methods are highly efficient for complex shape optimization problems.
New formula for portfolio risk management using conditional PDEs.
problem Optimal diversification and risk management of portfolios.
method Closed-form formula for conditional probability, Gaussian copulas, conditional risk-neutral PDE.
result Dynamic monitoring of portfolio volatilities and weights from PDEs.
Study uses outer metrics for PDE-constrained shape optimization over diffeomorphism group.
problem Optimizing shapes governed by PDEs over the diffeomorphism group.
method Outer metrics on diffeomorphism group, Riemannian steepest descent method.
result Riemannian approach outperforms other metrics in solving PDE-constrained shape optimization problems.
Paper studies deep learning for solving elliptic PDEs, proving optimal bounds and neural scaling laws.
problem Solving elliptic PDEs from random samples using machine learning.
method Deep Ritz Method and Physics-Informed Neural Networks (PINNs) for the Schrödinger equation.
result Proves minimax optimal bounds and neural scaling laws for deep PDE solvers.
We consider a specific type of nonlinear partial differential equations (PDE) that appear in mathematical finance as the result of solving some optimization problems. We review some existing in the literature examples of such problems, and discuss the properties of these PDEs. We also demonstrate how to solve them nume…
Novel method for shape optimization of non-smooth PDEs.
problem Optimizing shapes governed by non-smooth PDEs.
method Functional variational approach and sensitivity analysis.
result Necessary conditions for locally optimal shapes.
Paper tackles DOCTR-L with SciPhy RL, solving neural PDEs from data.
problem High-dimensional optimal control with stochastic policies.
method Soft HJB equation, Neural PDEs, Physics-Informed Neural Networks.
result Reduces DOCTR-L to solving neural PDEs from data.
In this paper we propose a new model-based unsupervised learning method, called VarNet, for the solution of partial differential equations (PDEs) using deep neural networks (NNs). Particularly, we propose a novel loss function that relies on the variational (integral) form of PDEs as apposed to their differential form …
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.
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.
AAS optimizes neural network PDE approximations by adaptively sampling.
problem Statistical errors from random samples in neural network PDE approximations.
method Minmax formulation to optimize neural network and training set samples.
result Reduces Monte Carlo approximation error for a given sample size.
The class of non-rigid registration methods proposed in the framework of PDE-constrained Large Deformation Diffeomorphic Metric Mapping is a particularly interesting family of physically meaningful diffeomorphic registration methods. PDE-constrained LDDMM methods are formulated as constrained variational problems, wher…
This article presents a new methodology called deep Theory of Functional Connections (TFC) that estimates the solutions of partial differential equations (PDEs) by combining neural networks with TFC. TFC is used to transform PDEs with boundary conditions into unconstrained optimization problems by embedding the boundar…
This paper develops a fast algorithm for solving nonlinear PDEs using sparse Cholesky factorization.
problem Efficiently solving nonlinear PDEs with Gaussian processes and kernel methods.
method Sparse Cholesky factorization for near-linear complexity.
result Near-linear complexity algorithm for working with kernel matrices of nonlinear PDEs.
Diffeologies unify infinite-dimensional geometry and PDEs, enhancing classical function spaces.
problem Combining infinite-dimensional geometry and PDEs for optimization problems.
method Review and extension of classical function spaces and mapping spaces.
result Diffeologies provide a unified framework for evolution equations and optimization problems.
New method uses neural networks to solve complex PDEs from optimal control theory.
problem Solving high-dimensional Hamilton-Jacobi-Bellman PDEs.
method Iterative diffusion optimization techniques, focusing on path measures and divergences.
result Favourable properties of log-variance divergence for Monte Carlo estimators.
New integration method improves BSDE-based PDE solvers.
problem Discretization bias in standard BSDE-based solvers.
method Proposed Stratonovich-based BSDE formulation with stochastic Heun integration.
result Eliminates bias issues and outperforms EM-based variants.
The paper develops a theory for one-step Wasserstein-guided models for PDE-induced measures.
problem Theoretical understanding of generative models' accuracy in scientific computing.
method Regularity theory for optimal transport between doubling measures, excess-risk bounds.
result One-step Wasserstein-guided generative models can approximate PDE-induced measures with Hölder continuity.
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.
A new method solves complex financial equations efficiently.
problem Solving worst-case and best-case prices for two-factor uncertain volatility models.
method Decompose and integrate, then optimize; piecewise constant control; closed-form Green's functions; 2D convolution integrals; monotone numerical integration; Fast Fourier Transforms.
result The method efficiently computes the value function and optimal control, converging to the viscosity solution of the HJB equation.
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.
Extracts coarse-grained PDEs from microscopic simulations.
problem Discovering effective PDEs for macro-scale processes from microscopic data.
method Combining neural networks with equation-free numerics and data-driven approaches.
result Efficiently discovers macro-scale PDEs from microscopic simulations.
Optimizes portfolios with constraints and stochastic factors, deriving explicit solutions.
problem Optimizing expected utility in an incomplete market with stochastic factors and convex constraints.
method Fundamental duality results and HJB PDE, derived condition for exponential affine solutions.
result Explicit expressions for optimal allocations and Riccati ODE solutions in specific markets.
New framework trains large SciML models solving PDEs in reasonable time.
problem Training large SciML models solving PDEs is challenging and time-consuming.
method Data parallel distributed deep learning framework with optimized methods.
result Neural PDE solvers can be viably trained for practical applications.
Bayesian PINNs optimize loss weights for PDEs and data.
problem Optimizing loss weights in physics-informed neural networks.
method Laplace approximation for efficient model evidence computation.
result Unified Bayesian setting for PDEs and noisy measurements.
New method detects changepoints in PDEs using optimized neural networks.
problem Detecting changepoints in PDEs with unknown locations and times.
method Online optimized Physics-Informed Neural Networks (PINNs) with Total-Variation penalty.
result Improved parameter estimation and model fitting with changepoints.
Bayesian PINNs learn elliptic PDEs with near-minimax posterior contraction rate.
problem Learning elliptic PDEs with noisy data and non-homogeneous boundary conditions.
method Bayesian approach with a Hölder space prior on neural network weights.
result Posterior contracts at near-minimax rate without prior knowledge of solution smoothness.
New methods improve deep learning for solving linear PDEs.
problem Efficiently solving high-dimensional linear PDEs using deep learning.
method Rigorous investigation of gradient estimators for SDE-based variational formulations.
result Novel methods provide substantial performance improvements.
A new machine learning method solves high-dimensional Kolmogorov PDEs efficiently.
problem Solving high-dimensional Kolmogorov PDEs and SDEs.
method Stochastic weighted minimization and stochastic gradient descent with Malliavin weights.
result Accurate approximation of high-dimensional Kolmogorov PDEs and SDEs without curse of dimensionality.
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.
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in such applications are high-dimensional as the dimension corresponds to the number …
The paper finds surfaces closest to being flat that span a given contour.
problem Finding surfaces in R3 that are as flat as possible while spanning a given contour. method The approach involves minimizing the total Gaussian curvature squared and solving a system of PDEs.
result The optimal surface is shown to be controlled by a biharmonic equation with specific boundary conditions.
Efficient surrogate modeling for complex PDEs with physical laws.
problem High computational cost of repeated PDE simulations.
method LC-prior Gaussian process with POD and RBF-FD.
result Significantly reduced computational cost and improved accuracy.
Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is embedded in a partial differential equation (PDE) that expresses the known physics and…
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.
Paper uses neural nets for financial optimization problems.
problem Financial optimization and derivative pricing problems.
method Neural networks and deep reinforcement learning for solving PDEs and dynamic optimization.
result Efficient resolution of nonlinear PDEs and dynamic optimization in finance.
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.
Discover equations from data using neural networks with constraints.
problem Discover equations from noisy data without theoretical derivation.
method Solve constrained optimization problem with penalty or trust-region barrier methods.
result Constrained method outperforms penalty method for higher noise levels or fewer collocation points.
Develops a new solver for path-dependent PDEs using signature kernels.
problem Solving path-dependent PDEs (PPDEs) efficiently and accurately.
method Uses signature kernels to solve PPDEs by approximating the solution with minimal norm in a reproducing kernel Hilbert space.
result Proves the consistency of the numerical scheme, ensuring convergence to PPDE solutions as the number of collocation points increases.
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.
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.
Gradient descent achieves optimal learning for elliptic PDEs via Sobolev norms.
problem Learning elliptic PDEs from noisy data.
method Gradient descent on Sobolev norm objective functions.
result Gradient descent achieves statistical optimality for elliptic PDEs.
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.
A new method combines classical and machine learning PDE solvers efficiently.
problem Combining classical and machine learning PDE solvers to reduce computational cost and improve accuracy.
method Proposes an approximate greedy router to select solvers at each iteration, mimicking a greedy approach.
result Consistently reduces final error and AUC of the error trajectory compared to single-solver baselines and hybrid approaches.
New algorithm optimizes nonlinear SDEs online with convergence guarantees.
problem Optimizing nonlinear stochastic differential equations (SDEs) is computationally challenging.
method Forward propagation algorithm that solves an SDE derived using forward differentiation.
result Convergence theorem for nonlinear dissipative SDEs with bounds on stochastic fluctuations.
Deep neural nets solve complex insurance math equations.
problem Optimal control problems in insurance math.
method Deep neural network algorithm for elliptic PDEs.
result Solves high-dimensional semilinear elliptic PDEs.