Probabilistic numerics expands numerical tasks with black box methods.
problem Difficult conditioning of random variables in numerical tasks.
method Construct probabilistic numerical methods based on final outputs, extrapolating limiting quantities.
result Higher orders of convergence achieved in various numerical tasks.
This paper tackles Bayesian system identification with probabilistic numerical methods.
problem Accurately modeling nonlinear dynamic systems from noisy data.
method Probabilistic Sequential Monte Carlo (SMC) combined with probabilistic numerical integration.
result Efficient identification of latent states and system parameters from noisy measurements.
We deliver a call to arms for probabilistic numerical methods: algorithms for numerical tasks, including linear algebra, integration, optimization and solving differential equations, that return uncertainties in their calculations. Such uncertainties, arising from the loss of precision induced by numerical calculation …
Parallel-in-time solver reduces ODE simulation time from linear to logarithmic.
problem Efficiently solving ordinary differential equations (ODEs) with reduced computational cost.
method Formulated a parallel-in-time probabilistic numerical ODE solver using time-parallel formulation of iterated extended Kalman smoothers.
result Reduces span cost from linear to logarithmic in the number of time steps.
Probabilistic solvers improve stability for stiff systems.
problem Performance penalties for small steps in stiff systems.
method Probabilistic exponential integrators that include fast linear dynamics in the prior.
result Proven L-stability and probabilistic error accounting.
New method stabilizes probabilistic ODE solvers for high accuracy.
problem Numerical instability in high-order ODE solvers.
method Accurate initialisation, coordinate change preconditioner, square-root implementation.
result Probabilistic ODE solvers can now achieve high order (up to 11) with stability.
New method combines ODE filters and numerical quadrature to propagate model uncertainty.
problem Propagation of model uncertainty in ODE solutions with uncertain parameters.
method Combining ODE filters with numerical quadrature.
result Effective propagation of both numerical and parametric uncertainty.
Probabilistic method combines space and time uncertainties in PDEs.
problem Separate treatment of space and time in PDE solvers obscures interactions and error quantification.
method Gaussian process interpretation of finite difference methods interacting with probabilistic ODE solvers.
result Joint quantification of space- and time-uncertainty possible without sacrificing ODE solver performance.
Prob-GParareal adds uncertainty quantification to PinT solvers for differential equations.
problem Uncertainty in numerical solutions of differential equations.
method Prob-GParareal uses Gaussian processes to model Parareal correction function, providing probabilistic forecasts.
result Prob-GParareal yields accurate and robust probabilistic forecasts on various ODE systems.
Developing stable and scalable probabilistic ODE solvers for stiff and high-dimensional problems.
problem Stiff and high-dimensional ODEs
method Matrix-free update step and iterative re-linearization
result Improved stability and scalability
The paper improves ODE solvers by integrating diverse information types.
problem Improving accuracy and physical meaningfulness of ODE solutions.
method Leveraging probabilistic solvers to include second-order information and physical conservation laws.
result Solutions become more accurate and physically meaningful with additional information.
A research frontier has emerged in scientific computation, wherein numerical error is regarded as a source of epistemic uncertainty that can be modelled. This raises several statistical challenges, including the design of statistical methods that enable the coherent propagation of probabilities through a (possibly dete…
Efficiently solves high-dimensional ODEs with probabilistic methods.
problem Solving high-dimensional ODEs with uncertainty quantification.
method Probabilistic numerical algorithm based on independence assumptions or Kronecker structure.
result Efficient probabilistic solutions for ODEs with millions of dimensions.
This article attempts to place the emergence of probabilistic numerics as a mathematical-statistical research field within its historical context and to explore how its gradual development can be related both to applications and to a modern formal treatment. We highlight in particular the parallel contributions of Sul'…
DALTON improves ODE parameter estimation by learning from noisy data.
problem High sensitivity to parameters in ODEs produces unreliable parameter estimates.
method Data-adaptive probabilistic likelihood approximation for ODEs.
result DALTON produces more accurate parameter estimates than existing methods.
Fenrir uses probabilistic numerics to simplify solving initial value problems.
problem Solving initial value problems in ordinary differential equations.
method Probabilistic numerics and Gauss--Markov regression.
result The method simplifies parameter estimation in ODEs, making it easier and more robust.
Deep learning improves probabilistic PPDE solution accuracy.
problem Approximating solutions to path-dependent PDEs with limited basis selection.
method Deep learning for conditional expectation estimation with error bounds.
result Deep learning yields more accurate PPDE solutions, especially in high dimensions.
We study a probabilistic numerical method for the solution of both boundary and initial value problems that returns a joint Gaussian process posterior over the solution. Such methods have concrete value in the statistics on Riemannian manifolds, where non-analytic ordinary differential equations are involved in virtual…
Calibrated probabilistic solvers improve accuracy of ODE estimates.
problem Uncertainty in probabilistic ODE solutions is not well-calibrated for adaptive step sizes.
method Introduce and assess several calibration methods for probabilistic ODE solvers.
result Calibration methods interact efficiently with adaptive step-size selection, improving posteriors.
New algorithm minimizes regret in multi-agent bandit problem with probabilistic communication.
problem Minimizing group regret in multi-agent multi-armed bandit problem with probabilistic communication.
method Proposes a new UCB-based algorithm for decentralized multi-agent multi-armed bandit problem on d-regular graphs with probabilistic communication. result The proposed algorithm outperforms state-of-the-art algorithms in minimizing group regret.
Scalable solver reduces PDE uncertainty with active learning.
problem High computational cost in solving PDEs.
method Stochastic dual descent and clustering-based active learning.
result Solver scales to large number of collocation points.
In this note, we extend an evolutionary stochastic portfolio optimization framework to include probabilistic constraints. Both the stochastic programming-based modeling environment as well as the evolutionary optimization environment are ideally suited for an integration of various types of probabilistic constraints. W…
We propose a novel sampling framework for inference in probabilistic models: an active learning approach that converges more quickly (in wall-clock time) than Markov chain Monte Carlo (MCMC) benchmarks. The central challenge in probabilistic inference is numerical integration, to average over ensembles of models or unk…
Novel probabilistic solver speeds up solving related linear systems.
problem Efficiently solving multiple related linear systems.
method Probabilistic linear solver over the parameter space, leveraging solved systems.
result Faster and more efficient solution of related linear systems.
Unified framework for differentiable graph partitioning with probabilistic cuts.
problem Lack of general guarantees and principled gradients in prior probabilistic relaxations of graph cuts.
method Unified probabilistic framework covering a wide class of cuts, including Normalized Cut, with tight analytic upper bounds.
result Rigorous, numerically stable foundation for scalable, differentiable graph partitioning.
Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic numerical methods that instead return a Gauss-Markov process defining a probability d…
There is renewed interest in formulating integration as an inference problem, motivated by obtaining a full distribution over numerical error that can be propagated through subsequent computation. Current methods, such as Bayesian Quadrature, demonstrate impressive empirical performance but lack theoretical analysis. A…
Improved probabilistic solar irradiance forecasting models for grid integration.
problem Enhancing accuracy of solar irradiance forecasts for grid integration.
method Developed and calibrated probabilistic models using post-hoc calibration techniques.
result NGBoost model with CRUDE calibration achieves comparable performance to numerical weather prediction models.
Model uses LLMs to process numerical data guided by natural language descriptions.
problem Challenges in integrating prior knowledge into probabilistic models.
method Developed LLM Processes to condition numerical predictive distributions on natural language.
result Improved predictive performance and structured qualitative descriptions.
This paper explores semi-qualitative probabilistic networks (SQPNs) that combine numeric and qualitative information. We first show that exact inferences with SQPNs are NPPP-Complete. We then show that existing qualitative relations in SQPNs (plus probabilistic logic and imprecise assessments) can be dealt effectively …
Probabilistic pseudo knots model uncertain knot diagrams.
problem Modeling knots with unresolved crossings.
method Assign probabilities to undetermined crossings; define probabilistic equivalence and extend classical knot invariants.
result Capture uncertainty in physical, biological, and computational contexts.
We examine doing probabilistic descent over manifolds implicitly defined by a set of polynomials with rational coefficients. The system of polynomials is assumed to be triangularized. An application of Whitney's embedding theorem allows us to work in a reduced dimensional embedding space. A numerical continuation metho…
SPPL simplifies probabilistic programming for exact inference.
problem Efficient exact inference in probabilistic models.
method SPPL translates probabilistic programs into sum-product expressions, leveraging new techniques for scalability.
result SPPL achieves up to 3500x speedups in exact inference.
This work frames active inference through control as inference, offering robust control algorithms.
problem Active inference framework lacks practical sensorimotor control algorithms.
method Frame active inference through control as inference, presenting trajectory optimization as inference.
result AI may be framed as partially-observed CaI when the cost function is defined in observation states.
Adaptive batch sizes improve active learning efficiency and flexibility.
problem Fixed batch sizes in active learning are inefficient due to dynamic cost-speed trade-offs.
method Probabilistic Numerics framework that adaptively changes batch sizes based on integration error and precision objectives.
result Significant enhancement in learning efficiency and flexibility across various applications.
Short-term load forecasting is a critical element of power systems energy management systems. In recent years, probabilistic load forecasting (PLF) has gained increased attention for its ability to provide uncertainty information that helps to improve the reliability and economics of system operation performances. This…
This study connects Gaussian processes and RKHS, bridging two machine learning communities.
problem Understanding the relationship between Gaussian processes and RKHS.
method Examining connections and equivalences in regression, interpolation, and other topics.
result Established the equivalence between Gaussian Hilbert space and RKHS.
A new standpoint on financial time series, without the use of any mathematical model and of probabilistic tools, yields not only a rigorous approach of trends and volatility, but also efficient calculations which were already successfully applied in automatic control and in signal processing. It is based on a theorem d…
We consider an original problem that arises from the issue of security analysis of a power system and that we name optimal discovery with probabilistic expert advice. We address it with an algorithm based on the optimistic paradigm and on the Good-Turing missing mass estimator. We prove two different regret bounds on t…
Solves probabilistic Lambert problem connecting astrodynamics with optimal mass transport.
problem Determining spacecraft velocity for given positions with probabilistic constraints.
method Generalized optimal mass transport (OMT) and Schrödinger bridge (SBP) connections.
result Existence and uniqueness of solution for probabilistic Lambert problem.
PPPD framework extracts physical characterizations from stochastic mechanical systems.
problem Complex system behavior requires more than probabilistic descriptions of QoI.
method Probabilistic Performance-Pattern Decomposition (PPPD) framework.
result Decomposes system behaviors into meaningful patterns in response space.
This paper describes and discusses Bayesian Neural Network (BNN). The paper showcases a few different applications of them for classification and regression problems. BNNs are comprised of a Probabilistic Model and a Neural Network. The intent of such a design is to combine the strengths of Neural Networks and Stochast…
The paper presents a machine learning framework to combine weather forecasts from multiple models.
problem Combining forecasts from different NWP models with varying biases and limitations.
method Three-stage framework using Quantile Regression Forests and quantile averaging.
result The framework generates well-calibrated probabilistic weather forecasts suitable for decision support.
New algorithm improves data imputation for complex multimodal data sets.
problem Artifacts in imputation methods for multimodal distributions.
method Combines kNN and KDE for probabilistic estimates. result Lower imputation errors and higher likelihood estimates.
Rigorous uncertainty quantification of probabilistic AI weather forecasts with conformal prediction
problem Calibrated uncertainty quantification in probabilistic weather forecasts
method Conformal prediction
result Calibrated uncertainty at no expense to other probabilistic metrics
Develops probabilistic safety regions for scalable classifiers.
problem Minimizing misclassification errors in supervised classification.
method Introduces probabilistic safety regions and scalable classifiers.
result Probabilistic certifications for classifier performance.
Simple probabilistic solution for optimal liquidation with linear price impact.
problem Maximizing expected terminal wealth in a setup with quadratic transaction costs.
method Provided a simple probabilistic solution to the problem.
result Simple and probabilistic form of the solution not previously published.
New DKPP family controls positive and negative dependence in random subsets.
problem Challenges in seamlessly bridging probabilistic models for positive and negative dependence.
method Introduced DKPP family and developed computational methods for probabilistic operations and inference.
result Controllability of positive and negative dependence demonstrated through numerical experiments.