Neural operators have emerged as promising surrogate models for solving partial differential equations (PDEs), but struggle to generalise beyond training distributions and are often constrained to a fixed temporal discretisation. This work introduces a physics-informed training framework that addresses these limitations by decomposing PDEs using operator splitting methods, training separate neural operators to learn individual non-linear physical operators while approximating linear operators with fixed finite-difference convolutions. This modular mixture-of-experts architecture enables generalisation to novel physical regimes by explicitly encoding the underlying operator structure. We formulate the modelling task as a neural ordinary differential equation (ODE) where these learned operators constitute the right-hand side, enabling continuous-in-time predictions through standard ODE solvers and implicitly enforcing PDE constraints. Demonstrated on incompressible and compressible Navier-Stokes equations, our approach achieves better convergence and superior performance when generalising to unseen physics. The method remains parameter-efficient, enabling temporal extrapolation beyond training horizons, and provides interpretable components whose behaviour can be verified against known physics.

Data-driven surrogate models offer fast, inexpensive approximations to complex numerical and experimental systems but typically lack uncertainty quantification, limiting their reliability in safety-critical applications. While Bayesian methods provide uncertainty estimates, they offer no statistical guarantees and struggle with high-dimensional spatio-temporal problems due to computational costs and dependence on prior specification. We present a conformal prediction (CP) framework that provides statistically guaranteed marginal coverage for surrogate models in a model-agnostic manner with near-zero computational cost. Our approach handles high-dimensional spatio-temporal outputs by performing cell-wise calibration while preserving the tensorial structure of predictions. Through extensive empirical evaluation across diverse applications-including partial differential equations, magnetohydrodynamics, weather forecasting, and fusion diagnostics-we demonstrate that CP achieves empirical coverage with valid error bars regardless of model architecture (Multi-layer perceptrons, U-Net, Fourier neural operator, ViT, GNN), training regime, or output dimensionality (spanning 32 to over 20 million dimensions). We evaluate three nonconformity scores (conformalised quantile regression, absolute error residual, and standard deviation) for both deterministic and probabilistic models, showing that guaranteed coverage holds even for out-of-distribution predictions where models are deployed on physics regimes different from their training data. Calibration requires only seconds to minutes on standard hardware, with prediction set construction incurring negligible computational overhead. The framework enables rigorous validation of pre-trained surrogate models for downstream applications without retraining, providing actionable uncertainty quantification for decision-making in scientific domains. While CP provides marginal rather than conditional coverage and assumes exchangeability between calibration and test data-limitations we demonstrate empirically through sensitivity analyses-our method circumvents the curse of dimensionality inherent in traditional uncertainty quantification approaches, offering a practical tool for the trustworthy deployment of machine learning in the physical sciences.

Predicting plasma evolution within a Tokamak is crucial to building a sustainable fusion reactor. Whether in the simulation space or within the experimental domain, the capability to forecast the spatio-temporal evolution of plasma field variables rapidly and accurately could improve active control methods on current tokamak devices and future fusion reactors. In this work, we demonstrate the utility of using Fourier Neural Operator (FNO) to model the plasma evolution in simulations and experiments. Our work shows that the FNO is capable of predicting magnetohydrodynamic models governing the plasma dynamics, 6 orders of magnitude faster than the traditional numerical solver, while maintaining considerable accuracy (NMSE $\sim 10^{-5})$. Our work also benchmarks the performance of the FNO against other standard surrogate models such as Conv-LSTM and U-Net and demonstrate that the FNO takes significantly less time to train, requires less parameters and outperforms other models. We extend the FNO approach to model the plasma evolution observed by the cameras positioned within the MAST spherical tokamak. We illustrate its capability in forecasting the formation of filaments within the plasma as well as the heat deposits. The FNO deployed to model the camera is capable of forecasting the full length of the plasma shot within half the time of the shot duration.

We combine reachability analysis and probability bounds analysis, which allow for imprecisely known random variables (multivariate intervals or p-boxes) to be specified as the initial states of a dynamical system. In combination, the methods allow for the temporal evolution of p-boxes to be rigorously computed, and they give interval probabilities for formal verification problems, also called failure probability calculations in reliability analysis. The methodology places no constraints on the input probability distribution or p-box and can handle dependencies generally in the form of copulas. We also provide a consonant approximation method for multivariate p-boxes, which allows for the prediction sets of dynamical systems to be efficiently computed. The presented methodology is rigorous and automatically verified, as both the dynamics and uncertainties are represented and solved with guaranteed enclosures.

Neural weather models have shown immense potential as inexpensive and accurate alternatives to physics-based models. However, most models trained to perform weather forecasting do not quantify the uncertainty associated with their forecasts. This limits the trust in the model and the usefulness of the forecasts. In this work we construct and formalise a conformal prediction framework as a post-processing method for estimating this uncertainty. The method is model-agnostic and gives calibrated error bounds for all variables, lead times and spatial locations. No modifications are required to the model and the computational cost is negligible compared to model training. We demonstrate the usefulness of the conformal prediction framework on a limited area neural weather model for the Nordic region. We further explore the advantages of the framework for deterministic and probabilistic models.

A finite-element method (FEM) model for the Mega-Ampere Spherical Tokamak - Upgrade (MAST-U) fusion tokamak has been developed to evaluate stress and deformations in the full device structure and to assess the stability of the whole tokamak with respect to its simulated exposure to an artificial level of neutron irradiation. Here, we use MAST-U as a proxy for a fusion power plant to explore the level of fidelity made possible by modern supercomputing systems. Gravity and atmospheric pressure were used to test the high-resolution FEM model, involving in excess of 122 million elements. Taking the MASTU fusion plasma as a neutron source, we perform full-scale neutron transport calculations to quantify spatial variations in the neutron flux and assess the neutron radiation exposure across the structure. This is a first step towards applying recently developed multiscale computational tools to evaluate the spectrum of stress in the tokamak, identifying the location of stress concentrations as well as their magnitude. This study provides an example of full fusion device neutronics and FEM simulations which are enabling UKAEA to define computational requirements for modelling a whole fusion power plant as well as for specifying operating conditions for the relevant materials.

Spherical tokamaks have many desirable features that make them an attractive choice for a future fusion power plant. Power-plant viability is intrinsically related to plasma heat and particle confinement, and this is often determined by the level of micro-instability-driven turbulence. Accurate calculation of the properties of turbulent microinstabilities is therefore critical for tokamak design; however, the evaluation of these properties is computationally expensive. The considerable number of geometric and thermodynamic parameters and the high resolutions required to accurately resolve these instabilities make repeated use of direct numerical simulations in integrated modeling workflows extremely computationally challenging and create the need for fast, accurate, reduced-order models. This paper outlines the development of a data-driven reduced-order model, often termed a surrogate model for the properties of micro-tearing modes (MTMs) across a spherical tokamak reactor-relevant parameter space utilizing Gaussian process regression and classification, techniques from machine learning. These two components are used in an active learning loop to maximize the efficiency of data acquisition, thus minimizing computational cost. The high-fidelity gyrokinetic code GS2 is used to calculate the linear properties of the MTMs: the mode growth rate, frequency, and normalized electron heat flux, and core components of a quasi-linear transport model. Data cross-validation and direct validation on unseen data are used to ascertain the performance of the resulting surrogate models.

The paper serves as a response to the recent challenge problem published by the NAFEMS Stochastic Working Group titled: "Uncertain Knowledge: A Challenge Problem" whereby the participants are to implement current practices and 'state-of-the-art' stochastic methods to address numerous uncertainty quantification problems presented in the challenge.In total, two different challenge problems on increasing complexity levels are addressed through the use of the following techniques: 1) Bayesian model updating for the calibration of the distribution models and model selection for the aleatory variables of interest; 2) Adaptivepinching method for the sensitivity analysis; and 3) Probability Bounds Analysis to quantify the uncertainty over the failure probabilities.For the reproducibility of the results and to provide a better understanding of the numerical techniques discussed in the paper, the MATLAB and R codes implemented to address the challenge problems are made available via: https://github.com/

We propose a correlated bivariate interval arithmetic which allows for an initial dependence to be propagated, as well as the tracking of complicated non-linear dependencies arising from a computer program’s execution. For this task, we extend several familiar concepts from probability theory to intervals, including bivariate copulas, conditioning, inference, and vine copulas. The interval copulas, which we call interval relations, may take any shape, and are represented by Boolean matrices defining where two intervals jointly exist or not. We use set conditioning to define an efficient correlated interval arithmetic, which may be used to find the input–output relations of operations. A key component of the presented arithmetic are interval relation networks, interval analogues to vine copulas, which store the interval relations throughout a program’s execution, and use set inference to determine any unknown relations. The presented network inference can give a robust outer approximation to the exact multivariate interval dependency, which is found by projecting each pairwise bivariate relation into higher dimensions. Although some higher dimensional information is lost in this process, the bivariate projections are often sufficient to stop interval bounds becoming excessively wide. This extension allows for intervals to be rigorously and tightly propagated in deterministic engineering codes in an automatic fashion, and we apply the arithmetic on several engineering dynamics problems, including a non-linear ordinary differential equation.

Preamble Verified numerical methods empower scientific computing to solve challenging mathematical problems whilst providing error guarantees. Differently from other methods, in verified methods, the error is a constituent component, it is contextually quantified and it can be reduced with more computational effort. These methods have been used to find rigorous—thus proven to be correct—solutions to notoriously difficult problems. An example of a challenging mathematical problem in engineering is the computation of the failure probability. Computing the failure probability of a complex system involves the solution of high dimensional integrals over a domain characterised by the preimage of a nonlinear function. Because of the complexity of this task, analytical rigour is sacrificed to find suitable numerical approximations. With verified methods the failure probability can be bounded numerically, while controlling the magnitude of the approximation. Motivations Bounding the failure probability of a complex system with the desired precision means being able to provide an interval where the reference solution resides with a given level of confidence. So far, only the Monte Carlo method has been used to provide reference values to failure probability problems whose analytical solution is not possible. Monte Carlo however, has a huge limitation: the desired precision cannot be reached on problems with very small target failure probability. Moreover, the efficiency of Monte Carlo deprecates when the problem is formulated to allow imprecise probability. Several adaptations of Monte Carlo have been proposed to allow for imprecise probability but none with error guarantees [1, 3]. Monte Carlo variations that allow for imprecision, e.g. second-order Monte Carlo,

In many practical situations, we know the probabilities a and b of two events A and B, and we want to estimate the joint probability Prob(A & B). The algorithm that estimates the joint probability based on the known values a and b is called an and-operation. An important case when such a reconstruction is possible is when we know the correlation between A and B; we call the resulting and-operation correlation-based. On the other hand, in statistics, there is a widely used class of and-operations known as copulas. Empirical evidence seems to indicate that the correlation-based and-operation derived in https://doi.org/10.1007/978-3-031-08971-8_64 is a copula, but until now, no proof of this statement was available. In this paper, we provide such a proof.

This work presents a new framework for uncertainty quantification developed as a package in the Julia programming language called UncertaintyQuantification.jl.Julia is a modern high-level dynamic programming language ideally suited for tasks like data analysis and scientific computing.UncertaintyQuantification.jl was developed from the ground up to be generalized and flexible while at the same time being easy to use.Leveraging the features of a modern language such as Julia allows to write efficient, fast and easy to read code.Especially noteworthy is Julia's core feature multiple dispatch which enables us to, for example, develop methods with a large number of varying simulation schemes such as standard Monte Carlo, Sobol sampling, Halton sampling, etc., yet minimal code duplication.Current features of UncertaintyQuantification.jl include simulation based reliability analysis using a large array of sampling schemes, local and global sensititivity analysis, meta modelling techniques such as response surface methodology or polynomial chaos expansion as well as the connection to external solvers by injecting values into plain text files as inputs.Through Julia's existing distributed computing capabilities all available methods can be easily run on existing clusters with just a few lines of extra code.

We present a method for the robust online updating of the parameters of a digital twin for engineering dynamics. The method is robust because it makes neither prior nor likelihood assumptions while rigorously quantifying the inferential uncertainty, which is useful in the context of scarce data. With the proposed updating strategy, the digital twin can update effectively instantaneously when presented with additional data. An algorithm for nonparametric inference with consonant random sets recently developed by the authors is applied. The method constructs a consonant joint structure of the updating parameters. This structure is composed of a sequence of nested sets, each of which is assigned a belief measure, i.e., a lower probability, a nominal level of confidence. This joint structure represents an outer approximation of the credal set where the target joint probability distribution resides. Thus, it can be regarded as a rigorous inferential model. The obtained joint structure has a possibilistic interpretation so that it can be seen as a joint fuzzy set and can be converted to a joint probability box using an established transformation. The method is arguably more efficient than traditional alternative Bayesian updating methods because it is intrinsically non-sequential thus parallelizable. The inference method can also be extended, with nearly no additional computational cost, to the case of interval measurement uncertainty. Because of its efficiency, the method can update the parameters of a digital twin model practically in real time, provided that (1) the data-generating mechanism is stationary, (2) the physical model has not changed, e.g. due to damage, and (3) input–output simulation data within the range of interest is available. We present an application to online updating of a digital twin of an aluminium three-storey structure subject to impact-hammer excitation.

We present a correlated and gate which may be used to propagate uncertainty and dependence through Boolean functions, since any Boolean function may be expressed as a combination of and and not operations. We argue that the and gate is a bivariate copula family, which has the interpretation of constructing bivariate Bernoulli random variables following a given Pearson correlation coefficient and marginal probabilities. We show how this copula family may be used to propagate uncertainty in the form of probabilities of events, probability intervals, and probability boxes, with only partial or no knowledge of the dependency between events, expressed as an interval for the correlation coefficient. These results generalise previous results by Fréchet on the conjunction of two events with unknown dependencies. We show an application propagating uncertainty through a fault tree for a pressure tank. This paper comes with an open-source Julia library for performing uncertainty logic.

The accuracy of Monte Carlo simulation methods depends on the computational effort invested in reducing the estimator variance.Typically reducing such variance requires invoking Monte Carlo with as many samples as one can afford.When the system is complex and the failure event is rare, it can be challenging to establish the correctness of the failure probability estimate.To combat this verification problem, we present an adaptation of the SIVIA algorithm (Set Inversion Via Interval Analysis) that computes rigorous bounds on the failure probability of rare events.With this method, the nonlinearity of the system and the magnitude of the failure event no longer constitute a limitation.This method can therefore be used for verification, when it is of interest to know the rigorous bounds of the very small target failure probability of complex systems, for example in benchmark problems.The method is rigorous i.e. inclusive and outside-in, so the more computational effort is invested the tighter the bounds.Because full separation is exercised between the engineering and the probability problem, the input uncertainty model can be changed without a re-evaluation of the physical function which opens avenues towards computing rigorous imprecise failure probability.For example, the reliability could be formulated without making dependency or distributional statements.

Elementary formulas for propagating information about means and variances through mathematical expressions have long been used by analysts. Yet the precise implications of such information are rarely articulated. This paper explores distribution-free techniques for risk analysis that do not require simulation, sampling or approximation of any kind. We describe best-possible bounds on risks that can be inferred given only information about the range, mean and variance of a random variable. These bounds generalise the classical Chebyshev inequality in an obvious way. We also collect in convenient tables several formulas for propagating range and moment information through calculations involving 7 binary convolutions (addition, subtraction, multiplication, division, powers, minimum, and maximum) and 9 unary transformations (scalar multiplication, scalar translation, exponentiation, natural and common logarithms, reciprocal, square, square root and absolute value) commonly encountered in risk expressions. These formulas are rigorous rather than approximate, and in most cases are either exact or mathematically best-possible. The formulas can be used effectively even when only interval estimates of the moments are available. Although most discussions of moment propagation assume stochastic independence among variables, this paper shows the assumption to be unnecessary and generalises formulas for the case when no assumptions are made about dependence, and when correlations are partially known. Along with partial means and variances, we show how interval covariances may be propagated and tracked through expressions.

Probability bounds analysis (PBA) is a collection of mathematical methods generalising interval analysis and probability theory. PBA can be utilised for uncertainty quantification for both aleatory and epistemic uncertainty across a wide range of scientific fields. PBA is most useful when information about variables is only partially known and can be used without requiring untenable assumptions to be made about parameter values, distribution shapes or dependence between variables. This paper introduces a PBA library for the Python programming language.

In this paper we present a framework for addressing a variety of engineering design challenges with limited empirical data and partial information. This framework includes guidance on the characterisation of a mixture of uncertainties, efficient methodologies to integrate data into design decisions, and to conduct reliability analysis, and risk/reliability based design optimisation. To demonstrate its efficacy, the framework has been applied to the NASA 2020 uncertainty quantification challenge. The results and discussion in the paper are with respect to this application.

[This corrects the article DOI: 10.1371/journal.pone.0240775.].

In this paper we perform nuclear data uncertain propagation with Total Monte Carlo, where the transport simulation is repeated for random evaluations of the data. The Oktavian Iron, Oktavian Nickel, and the Frascati Neutron Generator (FNG) neutron streaming SINBAD benchmarks were evaluated with OpenMC. Gaussian random deviates were drawn from the ENDF/B-VII.1 and TENDL-2017 libraries where the covariances were available. Uncertainty from multiple nuclides was propagated simultaneously assuming inter-nuclide independence. When the individual statistical uncertainty is negligible compared to the data uncertainty, then standard probability theory may be applied. If this is not the case and both need to be considered, we use Imprecise Probabilities (IP) to perform further analysis. We show how uncertain experimental data may be compared to uncertain simulation in the context of IP, and show how an uncertainty-based sensitivity analysis can be performed with IP.