Neural networks solve SPDEs using Wiener chaos expansion.
problem Solving stochastic partial differential equations (SPDEs) numerically.
method Using neural networks in the truncated Wiener chaos expansion.
result Approximation rates for learning SPDE solutions with noise.
The paper models asset prices using Wiener chaos expansions for efficient calibration to implied volatility surfaces.
problem Calibrating to implied volatility surfaces using flexible martingale models.
method Constructing an over-parameterized martingale model based on Wiener chaos expansions and conditional expectations.
result The method enables fast calibration to implied volatility surfaces and demonstrates flexibility through numerical experiments.
This paper proposes a new method to optimize portfolio allocation with transaction costs using Wiener chaos expansion.
problem Optimizing portfolio allocation with transaction costs in multi-period settings.
method Wiener chaos expansion approach to represent and solve the optimization problem.
result The proposed method finds an optimal strategy for portfolio allocation with transaction costs.
In this work, we propose a new policy iteration algorithm for pricing Bermudan options when the payoff process cannot be written as a function of a lifted Markov process. Our approach is based on a modification of the well-known Longstaff Schwartz algorithm, in which we basically replace the standard least square regre…
New method estimates SDE parameters efficiently using WCE and SGD.
problem Parameter estimation for stochastic differential equations.
method Wiener Chaos Expansion and Stochastic Gradient Descent.
result Accurate parameter recovery from noisy observations.
New method distinguishes data noise from GP uncertainty.
problem Uncertainty in kernel regression with non-Gaussian noise.
method Wiener chaos expansions for non-Gaussian noise.
result Can distinguish aleatoric from epistemic uncertainty.
In this we paper we recast the Cox--Ingersoll--Ross model of interest rates into the chaotic representation recently introduced by Hughston and Rafailidis. Beginning with the ``squared Gaussian representation'' of the CIR model, we find a simple expression for the fundamental random variable X. By use of techniques fro…
This work explores functional expansions to handle path dependence in various fields.
problem Path dependence and infinite-dimensional problems in non-Markovian systems.
method Generalizes Wiener series and functional Taylor expansion to handle static and dynamic functionals.
result Elegant separation of functionals from future trajectories in dynamic cases.
The Wiener chaos approach to interest rate modelling arises from the observation that the pricing kernel admits a representation in terms of the conditional variance of a square-integrable random variable, which in turn admits a chaos expansion. When the expansion coefficients factorise into multiple copies of a single…
Study reveals three limiting regimes for neural network functionals.
problem Understanding the behavior of functionals of random neural networks.
method Central and non-central limit theorems, Hermite expansions, Diagram Formula, Stein-Malliavin techniques.
result Three distinct limiting regimes based on fixed points of covariance function.
Study on spin random fields using chaos decomposition for cosmic microwave background modeling.
problem Modeling polarization of Cosmic Microwave Background using spin random fields.
method Explicit Wiener-Itô chaos decomposition of area measures of level sets.
result Reveals a clear difference between high frequency regime and zero spin case.
In this work, we propose an algorithm to price American options by directly solving the dual minimization problem introduced by Rogers. Our approach relies on approximating the set of uniformly square integrable martingales by a finite dimensional Wiener chaos expansion. Then, we use a sample average approximation tech…
In the setting proposed by Hughston & Rafailidis (2005) we consider general interest rate models in the case of a Brownian market information filtration (Ft)t≥0. Let X be a square-integrable F∞-measurable random variable, and assume the non-degeneracy condition that for all $t<\in…
Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.
problem Failure of Gaussian equivalence in polynomial feature embeddings under quadratic scaling.
method Introduced Conditional Gaussian Equivalent (CGE) model to capture non-Gaussian behavior.
result Correct asymptotics derived for training and test errors in CGE model.
New chaos formula simplifies variance calculation for Gaussian nodal volumes.
problem Analyzing the variance of Gaussian nodal volumes on Riemannian manifolds.
method Explicit Wiener-Itô chaos decomposition, reducing complexity from 2+2n to 4 Hermite polynomials. result New exact formula for variance and bounds, valid for arbitrary manifolds.
In an abstract Wiener space setting, we constract a rigorous mathematical model of the one-loop approximation of the perturbative Chern-Simons integral, and derive its explicit asymptotic expansion for stochastic Wilson lines.
Enhances polynomial chaos models with uncertainty intervals.
problem Uncertainty quantification in surrogate models.
method Jackknife-based conformal prediction integrated into polynomial chaos expansions.
result Produces accurate predictive intervals for low-accuracy models.
Sparse Polynomial Chaos expansions improve accuracy and efficiency in simulations.
problem Challenges in computational efficiency and accuracy for Polynomial Chaos modeling.
method Sparse Bayesian learning using Variational Relevance Vector Machines.
result Sparse Polynomial Chaos expansions achieve comparable performance to compressive sensing with fewer data points.
Gradient-enhanced GSA uses Poincaré chaos expansions for accurate sensitivity analysis.
problem Accurately estimating Sobol' indices with limited data.
method Integrates sparse, gradient-enhanced regression with Poincaré chaos expansions for derivative-based sensitivity analysis.
result Accurately estimated Sobol' indices using limited data.
A new method builds sparse polynomial chaos expansions for models with dependent inputs.
problem Quantifying uncertainty in models with dependent inputs.
method Data-driven approach to construct orthonormal polynomials recursively based on input correlations.
result Reduces the number of observations and improves numerical stability and computational efficiency.
Basis adaptation in Homogeneous Chaos spaces rely on a suitable rotation of the underlying Gaussian germ. Several rotations have been proposed in the literature resulting in adaptations with different convergence properties. In this paper we present a new adaptation mechanism that builds on compressive sensing algorith…
Bayesian adaptive PCE method improves surrogate modeling and sensitivity analysis.
problem Lack of fully Bayesian PCE methods in statistics.
method Developed a novel fully Bayesian adaptive PCE method with R implementation.
result Bayesian adaptive PCE provides competitive performance for various UQ tasks.
Polynomial Chaos Expansion improves operator learning for PDEs.
problem Approximating mappings between infinite-dimensional functional spaces.
method Polynomial Chaos Expansion (PCE) for operator learning.
result PCE achieves strong performance in operator learning and uncertainty quantification.
Combines Gaussian processes and polynomial chaos for stochastic control.
problem Uncertainties in dynamic models lead to performance issues in predictive control.
method Combines Gaussian processes with polynomial chaos expansions to estimate probability distributions of nonlinear functions.
result Demonstrates accurate approximation and closed-loop performance in stochastic nonlinear model predictive control.
NeuralChaos efficiently approximates complex stochastic processes.
problem Representing and computing square-integrable predictable processes over time.
method Introduces NeuralChaos, a neural operator architecture for Rd-valued predictable processes. result NeuralChaos achieves best N-term chaoslet approximation rates and is dense in HT2(Rd). This paper simplifies conditional Sobol' indices calculation using PCE bases.
problem Computational inefficiency and lack of consistency in evaluating conditional Sobol' indices.
method Analytical extraction of conditional Sobol' indices via basis decomposition of PCE expansions.
result Derives closed-form expressions for conditional Sobol' indices.
AL-SPCE improves reliability analysis for complex systems with active learning and SPCE.
problem Efficiently analyzing reliability of complex, computationally expensive models with intrinsic randomness.
method Active learning framework using stochastic polynomial chaos expansions (SPCE) to reduce computational burden.
result AL-SPCE maintains high accuracy in reliability estimates while significantly improving efficiency.
Enhanced PC2 improves surrogate modeling for high-dimensional problems.
problem Degrading performance and efficiency of PC2 in high-dimensional parameter spaces. method Integrates SULM solver and D-optimal sampling strategy into PC2 framework. result Enhanced PC2 demonstrates better comprehensive capability and efficiency. Model reduction methods aim to describe complex dynamic phenomena using only relevant dynamical variables, decreasing computational cost, and potentially highlighting key dynamical mechanisms. In the absence of special dynamical features such as scale separation or symmetries, the time evolution of these variables typi…
Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.
problem Uncertainty quantification in high-dimensional stochastic systems.
method Principal Geodesic Analysis on the Grassmann manifold, adaptive algorithm for local submanifolds, polynomial chaos expansion.
result Efficient surrogate modeling of system behavior across different parameter spaces.
A new method for creating simpler models from complex ones.
problem Creating accurate approximations of complex models at reduced costs.
method Sequential adaptive surrogate modeling based on locally spectral expansions.
result Stochastic spectral embedding (SSE) shows good approximation capabilities and scalability.
Paper uses PCE to quantify ML model and input uncertainties.
problem Accurately quantify and propagate combined uncertainties in ML predictions.
method Polynomial Chaos Expansion (PCE) for joint input and model uncertainty.
result Efficient and accurate calculation of output variability and sensitivity.
Enhances PCE surrogates using transfer learning for expensive simulations.
problem Over-sampling in PCE for expensive forward models.
method Transfer learning from similar tasks to a new task with limited training data.
result Improves scalability and accuracy of PCE surrogates.
A new method combines SciML and UQ with physical constraints.
problem Uncertainty quantification in scientific machine learning tasks.
method Physics-constrained polynomial chaos expansion.
result Effective uncertainty quantification and SciML integration.
We develop a technique based on Malliavin-Bismut calculus ideas, for asymptotic expansion of dual control problems arising in connection with exponential indifference valuation of claims, and with minimisation of relative entropy, in incomplete markets. The problems involve optimisation of a functional of Brownian path…
Polynomial chaos surrogates handle intrinsic noise in stochastic models.
problem Handling intrinsic noise in stochastic models with parametric uncertainty.
method Developed a PCE surrogate on a joint space of intrinsic and parametric uncertainty using Rosenblatt transformations and Karhunen-Loeve expansion.
result Quantified intrinsic noise contribution to model output variance using PCE Sobol indices.
Bayesian optimization with RPCE reduces MAP estimation for structural dynamics models.
problem Estimating parameters of structural dynamic models efficiently.
method Bayesian optimization with RPCE surrogate model.
result Effective reduction in model evaluations for MAP estimation.
CODE learns ODE dynamics from sparse data, outperforming neural and kernel methods.
problem Learning ODE dynamics from sparse and noisy data.
method CODE uses Polynomial Chaos Expansion (aPCE) for the ODE's RHS, enabling global orthonormal polynomial representation.
result CODE exhibits remarkable extrapolation capabilities even under novel initial conditions and measurement noise.
Conformal prediction improves prediction intervals for PCEs, especially in sparse cases.
problem Quantifying local model errors in PCEs for small datasets.
method Integration of conformal prediction methods (full and Jackknife+) into full and sparse PCEs.
result Better-calibrated prediction intervals for both full and sparse PCEs.
Surrogate model construction for vector-valued outputs
problem Improving surrogate model accuracy and stability for complex engineering systems
method Adaptive sequential sampling for polynomial chaos expansion
result Improves surrogate accuracy and stability
Neural Chaos uses neural networks instead of polynomials for stochastic modeling.
problem Challenges in constructing surrogate models with uncertainty quantification for complex or high-dimensional stochastic processes.
method Adopting spectral expansion formalism with neural network basis functions, identifying them data-drivenly without prior assumptions.
result Demonstrates effectiveness of the proposed scheme through numerical examples of varying complexity.
This paper optimizes PCE for efficient surrogate modeling in engineering.
problem Efficiently selecting polynomial regressors for surrogate modeling in computationally expensive models.
method Three state-of-the-art basis-adaptive sparse PCE methods are compared and analyzed.
result Automatic selection of the best solver and basis-adaptive scheme improves surrogate model accuracy.
A new neural network model uses polynomial chaos theory to improve neural signal processing.
problem Redundant neural signal representation in DANNs.
method Employing arbitrary polynomial chaos theory to construct orthonormal representations in DANNs.
result Improves neural signal processing by reducing redundancy and enhancing orthogonality.
Bayesian approach improves sparse PCE for high-dimensional problems.
problem Sparse PCE struggles with high-dimensional uncertainty and underdetermined situations.
method Joint shrinkage priors and MCMC for sparse PCE with uncertainty estimation.
result Bayesian PCE achieves sparse representations with higher polynomial degrees.
We present a regression technique for data-driven problems based on polynomial chaos expansion (PCE). PCE is a popular technique in the field of uncertainty quantification (UQ), where it is typically used to replace a runnable but expensive computational model subject to random inputs with an inexpensive-to-evaluate po…
PCENet reduces uncertainty in high-dimensional data efficiently.
problem Uncertainty quantification in high-dimensional data is computationally expensive.
method Two-stage learning process: variational autoencoder for low-dimensional representation, polynomial chaos expansion for mapping.
result Model captures system dynamics, learns under uncertainty, estimates high-dimensional data uncertainty, matches output distribution moments.
We consider a Markov process X, which is the solution of a stochastic differential equation driven by a Lévy process Z and an independent Wiener process W. Under some regularity conditions, including non-degeneracy of the diffusive and jump components of the process as well as smoothness of the Lévy density of $Z…
We compare the option pricing formulas of Louis Bachelier and Black-Merton-Scholes and observe -- theoretically as well as for Bachelier's original data -- that the prices coincide very well. We illustrate Louis Bachelier's efforts to obtain applicable formulas for option pricing in pre-computer time. Furthermore we ex…