Model Form Error Research Articles

Investigating Model Form Error Estimation for Sparse Data

Computational simulations of dynamical systems involve the use of mathematical models and algorithms to mimic and analyze complex real-world phenomena. By leveraging computational power, simulations enable researchers to explore and understand systems that are otherwise challenging to study experimentally. They offer a cost-effective and efficient means to predict and analyze the behavior of physical, biological, and social systems. However, model form error arises in computational simulations from simplifications, assumptions, and limitations inherent in the mathematical model formulation. Several methods for addressing model form error have been proposed in the literature, but their robustness in the face of challenges inherent to real-world systems has not been thoroughly investigated. In this work, a data assimilation-based approach for model form error estimation is investigated in the presence of sparse observation data. Including physics-based domain knowledge to improve estimation performance is also explored. The Lotka-Volterra equations are employed as a simple computational simulation for demonstration.

Generalized field inversion strategies for data-driven turbulence closure modeling

Most data-driven turbulence closures are based on the general structure of nonlinear eddy viscosity models. Although this structure can be embedded into the machine learning algorithm and the Reynolds stress tensor itself can be fit as a function of scalar- and tensor-valued inputs, there exists an alternative two-step approach. First, the spatial distributions of the optimal closure coefficients are computed by solving an inverse problem. Subsequently, these are expressed as functions of solely scalar-valued invariants of the flow field by virtue of an arbitrary regression algorithm. In this paper, we present two general inversion strategies that overcome the limitation of being applicable only when all closure tensors are linearly independent. We propose to either cast the inversion into a constrained and regularized optimization problem or project the anisotropy tensor onto a set of previously orthogonalized closure tensors. Using the two-step approach together with either of these strategies then enables us to quantify the model-form error associated with the closure structure independent of a particular regression algorithm. Eventually, this allows for the selection of the a priori optimal set of closure tensors for a given, arbitrary complex test case.

An information field theory approach to Bayesian state and parameter estimation in dynamical systems

Dynamical system state estimation and parameter calibration problems are ubiquitous across science and engineering. Bayesian approaches to the problem are the gold standard as they allow for the quantification of uncertainties and enable the seamless fusion of different experimental modalities. When the dynamics are discrete and stochastic, one may employ powerful techniques such as Kalman, particle, or variational filters. Practitioners commonly apply these methods to continuous-time, deterministic dynamical systems after discretizing the dynamics and introducing fictitious transition probabilities. However, approaches based on time-discretization suffer from the curse of dimensionality since the number of random variables grows linearly with the number of time-steps. Furthermore, the introduction of fictitious transition probabilities is an unsatisfactory solution because it increases the number of model parameters and may lead to inference bias. To address these drawbacks, the objective of this paper is to develop a scalable Bayesian approach to state and parameter estimation suitable for continuous-time, deterministic dynamical systems. Our methodology builds upon information field theory. Specifically, we parameterize the dynamical system state path using a linear basis. Then, we construct a physics-informed prior probability measure for the coefficients of the linear basis such that functions that satisfy the physics are more likely. This prior allows us to quantify model form errors. We connect the system's response to observations through a probabilistic model of the measurement process. The joint posterior over the system responses and all parameters is given by Bayes' rule. To approximate the intractable posterior, we develop a stochastic variational inference algorithm. In summary, the developed methodology offers a powerful framework for Bayesian estimation in dynamical systems.

A data-driven multiscale model for reactive wetting simulations

We describe a data-driven, multiscale technique to model reactive wetting of a silver–aluminum alloy on a Kovar™ (Fe-Ni-Co alloy) surface. We employ molecular dynamics simulations to elucidate the dependence of surface tension and wetting angle on the drop’s composition and temperature. A design of computational experiments is used to efficiently generate training data of surface tension and wetting angle from a limited number of molecular dynamics simulations. The simulation results are used to parameterize models of the material’s wetting properties and compute the uncertainty in the models due to limited data. The data-driven models are incorporated into an engineering-scale (continuum) model of a silver–aluminum sessile drop on a Kovar™ substrate. Model predictions of the wetting angle are compared with experiments of pure silver spreading on Kovar™ to quantify the model-form errors introduced by the limited training data versus the simplifications inherent in the molecular dynamics simulations. The paper presents innovations in the determination of “convergence” of noisy MD simulations before they are used to extract the wetting angle and surface tension, and the construction of their models which approximate physio-chemical processes that are left unresolved by the engineering-scale model. Together, these constitute a multiscale approach that integrates molecular-scale information into continuum scale models.

Analytic Error Analysis of the Partial Derivatives Cross-Section Model—I: Derivation

Accurate assessment of uncertainties in cross-section data is crucial for reliable nuclear reactor simulations and safety analyses. In this study, we focus on the interpolation procedure of the partial derivatives (PD) cross-section model used to evaluate nodal parameters from pregenerated multigroup libraries. Our primary objective is to develop a systematic methodology that enables prediction of the incurred errors in the cross-section model, leading to the development of optimal case matrices, more efficient cross-section models, and informed case matrix construction for reactor types lacking substantial data and experience. We make progress toward this objective through a rigorous analytic error analysis enabled by the derivation of error expressions and bounds for the PD model based on the discovery that the method is a form of Lagrange interpolation. Our investigations reveal distinct outcomes depending on the chosen cross-section functionalizations, achieved by identifying the sources of error. These error sources are found to include interpolation error, which is always present, and model form error, which is a property of the supplied case matrix. We show that simply increasing grid refinement through the addition of branches may not always lead to decreased cross-section errors, particularly in cases where the model form error predominantly contributes to the total error. We present numerical results and verification in a companion paper, showing these different error characteristics for various cross-section functionalizations. Although applied to current light water reactor environments, our methodology offers a means for advanced reactor analysts to develop case matrices with quantified error levels, advancing the goal of a general methodology for robust two-step reactor analysis. Future work includes exploring different lattice types and functionalizations, extending reactivity bounds to multilattice problems, and investigating historical effects within the macroscopic depletion model.

Stochastic model corrections for reduced Lotka-Volterra models exhibiting mutual, commensal, competitive, and predatory interactions.

We explore model-form error and how to correct it in systems of ordinary differential equations. In particular, we focus on the Lotka-Volterra equations, which are used broadly in fields such as ecology, biology, economics, chemistry, and physics. Accounting for every object and their complex interactions with a complete model often becomes infeasible, thereby requiring reduced models. However, reduced models may omit vital relationships, resulting in discrepancies between reduced model predictions and observations from the true system. In this work, we propose a model correction framework for decreasing such discrepancies. Specifically, we embed a stochastic enrichment operator into the reduced model's system of equations. The enrichment operator is theory-informed, calibrated with observations from the complete model, and extended to extrapolative combinations of parameters and initial conditions. The complete model involves N species, while the reduced and enriched models only track M<N species. Numerical results show the enriched models significantly decrease discrepancies, consistently predict equilibria, and improve the species' transient behavior.

Safeguarding Multi-Fidelity Bayesian Optimization Against Large Model Form Errors and Heterogeneous Noise

Abstract Bayesian optimization (BO) is a sequential optimization strategy that is increasingly employed in a wide range of areas such as materials design. In real-world applications, acquiring high-fidelity (HF) data through physical experiments or HF simulations is the major cost component of BO. To alleviate this bottleneck, multi-fidelity (MF) methods are used to forgo the sole reliance on the expensive HF data and reduce the sampling costs by querying inexpensive low-fidelity (LF) sources whose data are correlated with HF samples. However, existing multi-fidelity BO (MFBO) methods operate under the following two assumptions that rarely hold in practical applications: (1) LF sources provide data that are well correlated with the HF data on a global scale, and (2) a single random process can model the noise in the MF data. These assumptions dramatically reduce the performance of MFBO when LF sources are only locally correlated with the HF source or when the noise variance varies across the data sources. In this paper, we view these two limitations and uncertainty sources and address them by building an emulator that more accurately quantifies uncertainties. Specifically, our emulator (1) learns a separate noise model for each data source, and (2) leverages strictly proper scoring rules in regularizing itself. We illustrate the performance of our method through analytical examples and engineering problems in materials design. The comparative studies indicate that our MFBO method outperforms existing technologies, provides interpretable results, and can leverage LF sources which are only locally correlated with the HF source.

Flow enhancement from wall pressure observations: A compressible continuous adjoint data assimilation model

This study establishes a compressible continuous adjoint data assimilation (C2ADA) approach for reproducing a complete mean flow from sparse wall pressure observations. The model-form error induced by the Boussinesq approximation is corrected by the addition of a spatially varying additive forcing term. The linear part of the eddy viscosity, computed using the conventional Reynolds-averaged Navier–Stokes model, is incorporated for ensuring the well-posedness of the optimization. The model is derived theoretically to minimize discrepancies between the wall pressure measurements and the numerical predictions of the primary-adjoint system, thereby enabling determination of the optimal contribution of the Reynolds force vector. The effects of divergence schemes and turbulence models are investigated by examining flow over a 30P30N airfoil. The C2ADA model, employing two distinct schemes, demonstrates significant improvements in velocity estimation, but the first-order scheme introduces excessive dissipation, resulting in an under-prediction of spanwise vorticity. The C2ADA model combined with different eddy-viscosity models uniquely recovers the Reynolds force vectors and obtains mean fields that outperform those achieved solely through conventional eddy viscosity models. The practicability of the C2ADA model for capturing complex flow phenomena is confirmed by applying it to study three-dimensional flow over a 65° delta wing. Despite limited wall pressure observations, the C2ADA model has shown a notable improvement in accurately estimating the intensity and location of both the primary and secondary vortices. Recovery errors in the apex region are significantly diminished by incorporating a paucity of observations account for the effect of inboard vortex. The study broadens the applicability of continuous adjoint-based approaches for modeling compressible flow, as our C2ADA approach is easily implemented in existing computational fluid dynamics solvers and has significantly higher computational efficiency than other approaches.

Dimensionality reduction for regularization of sparse data-driven RANS simulations

Data assimilation can reduce the model-form errors of RANS simulations. A spatially distributed corrective parameter field can be introduced into the closure model, whose optimal values can be efficiently found by an adjoint method and gradient-based optimization. When assimilating experimental data, which in most cases are sparsely distributed or based on a low-resolution grid, the inverse problem is highly underdetermined and thus ambiguous. Therefore, to reduce the ambiguity, regularization is required. Established regularization approaches such as total variation and Sobolev gradient methods can produce smooth and physically meaningful velocity fields in many cases. However, if the measurements are located close to walls, spiky and unphysical wall shear stress profiles can occur. A new regularization strategy based on a piecewise linear approximation of the corrective field is proposed. This method is shown to lead to a very accurate free stream velocity field and smooth wall shear stress profiles. The resulting skin friction drag error for the case of flow over periodic hills was around 1.38% which is seven times lower than the error obtained with the Sobolev gradient and two orders of magnitude lower than that obtained with the other two methods.

Physics-informed information field theory for modeling physical systems with uncertainty quantification

Data-driven approaches coupled with physical knowledge are powerful techniques to model engineering systems. The goal of such models is to efficiently solve for the underlying physical field through combining measurements with known physical laws. As many physical systems contain unknown elements, such as missing parameters, noisy measurements, or incomplete physical laws, this is widely approached as an uncertainty quantification problem. The common techniques to handle all of these variables typically depend on the specific numerical scheme used to approximate the posterior, and it is desirable to have a method which is independent of any such discretization. Information field theory (IFT) provides the tools necessary to perform statistics over fields that are not necessarily Gaussian. The objective of this paper is to extend IFT to physics-informed information field theory (PIFT) by encoding the functional priors with information about the physical laws which describe the field. The posteriors derived from this PIFT remain independent of any numerical scheme, and can capture multiple modes which allows for the solution of problems which are not well-posed. We demonstrate our approach through an analytical example involving the Klein-Gordon equation. We then develop a variant of stochastic gradient Langevin dynamics to draw samples from the field posterior and from the posterior of any model parameters. We apply our method to several numerical examples with various degrees of model-form error and to inverse problems involving non-linear differential equations. As an addendum, the method is equipped with a metric which allows the posterior to automatically quantify model-form uncertainty. Because of this, our numerical experiments show that the method remains robust to even an incorrect representation of the physics given sufficient data. We numerically demonstrate that the method correctly identifies when the physics cannot be trusted, in which case it automatically treats learning the field as a regression problem.

Turbulence model form errors in separated flows

Physics-based uncertainty quantification is applied to a boundary layer over a flat plate with a statistically stationary separation bubble to assess sources and dynamics of model error (ME) in two-equation Reynolds-Averaged Navier-Stokes turbulence models in separated flows. Two distinct ME modes are found that correspond to the qualitative behavior of ME in a turbulent wall-bounded flow (upstream of the separation bubble) and turbulent free-shear flow (within the separation bubble), with a superposition of these ME modes in intermediate regions. A complex understanding of ME results as it changes throughout the flow, but is ultimately comprised of the elements of canonical flows.

Application of the p-version of FEM to hierarchic rod models with reference to mechanical contact problems

The formulation of a system of hierarchic models for the simulation of the mechanical response of slender elastic bodies, such as elastic rods, is considered. The present work is concerned with aspects of implementation and numerical examples. We use a finite element formulation based on the principle of minimum potential energy. The displacement fields are represented by the product of one-dimensional field functions and two-dimensional director functions. The field functions are approximated by the p-version of the finite element method. Our objective is to control both the model form errors and the errors of discretization with a view toward the development of advanced engineering applications equipped with autonomous error control procedures. We present numerical examples that illustrate the performance characteristics of the algorithm.

Probabilistic physics-informed machine learning for dynamic systems

This paper develops a physics-informed machine learning approach for response prediction in dynamic systems, by augmenting a physics-based model with a machine learning model for model error; both models are probabilistic. The physics-based dynamic system model has discrepancy in the predicted output (due to incomplete physics or model form error in the physics-based model) when compared against observation data. The model form error is quantified using Bayesian state estimation with observation data, and a probabilistic machine learning model is trained to predict the output discrepancy for untrained inputs. Two different computational options are developed to implement this overall approach, with different levels of sophistication and computational effort. The methods are developed for problems with stationary and non-stationary, and Gaussian and non-Gaussian random process inputs. Strategies for computational effort reduction at different stages, such as model training, model error quantification, and reliability prediction, are discussed. The proposed approach is demonstrated using two numerical illustrations: (a) a deep beam subjected to dynamic loading (a single physics system), and (b) hypersonic flow over a flexible aircraft panel (a multi-physics system).

Calibrating hypersonic turbulence flow models with the HIFiRE-1 experiment using data-driven machine-learned models

In this paper we study the efficacy of combining machine-learning methods with projection-based model reduction techniques for creating data-driven surrogate models of computationally expensive, high-fidelity physics models. Such surrogate models are essential for many-query applications e.g., engineering design optimization and parameter estimation, where it is necessary to invoke the high-fidelity model sequentially, many times. Surrogate models are usually constructed for individual scalar quantities. However there are scenarios where a spatially varying field needs to be modeled as a function of the model’s input parameters. We develop a method to do so, using projections to represent spatial variability while a machine-learned model captures the dependence of the model’s response on the inputs. The method is demonstrated on modeling the heat flux and pressure on the surface of the HIFiRE-1 geometry in a Mach 7.16 turbulent flow. The surrogate model is then used to perform Bayesian estimation of freestream conditions and parameters of the SST (Shear Stress Transport) turbulence model embedded in the high-fidelity (Reynolds-Averaged Navier–Stokes) flow simulator, using shock-tunnel data. The paper provides the first-ever Bayesian calibration of a turbulence model for complex hypersonic turbulent flows. We find that the primary issues in estimating the SST model parameters are the limited information content of the heat flux and pressure measurements and the large model-form error encountered in a certain part of the flow.

A Perspective on Data-Driven Coarse Grid Modeling for System Level Thermal Hydraulics

In the future, advanced reactors are expected to play an important role in nuclear power. However, their development and deployment are hindered by the absence of reliable and efficient models for analysis of system thermal hydraulics (TH). For instance, mixing and thermal stratification in reactor enclosures cannot be captured by traditional one-dimensional system codes, yet usage of high-resolution solvers is computationally expensive. Recent developments of coarse grid (CG) and system codes with three-dimensional capabilities have shown that they are promising tools for system-level analysis. However, these codes feature large turbulence model form and discretization errors and require further improvements to capture turbulent effects during complex transients. Improvements can be achieved by using data-driven (DD) approaches. This paper provides an overview of recent applications of DD methods in the areas of fluid dynamics and TH. It is demonstrated that they are being widely applied for engineering-scale analysis (e.g., closures for large eddy simulations/Reynolds-averaged Navier-Stokes using direct numerical simulation data). However, they cannot be directly employed for the system scale because of some features of the latter: usage of CG, transient nature of the considered phenomena, nonlinear interaction of multiple closures, etc. At the same time, accumulated experience allows outlining of potential frameworks for further developments in DD CG modeling of system-level TH.

Hybrid quadrature moment method for accurate and stable representation of non-Gaussian processes applied to bubble dynamics.

Solving the population balance equation (PBE) for the dynamics of a dispersed phase coupled to a continuous fluid is expensive. Still, one can reduce the cost by representing the evolving particle density function in terms of its moments. In particular, quadrature-based moment methods (QBMMs) invert these moments with a quadrature rule, approximating the required statistics. QBMMs have been shown to accurately model sprays and soot with a relatively compact set of moments. However, significantly non-Gaussian processes such as bubble dynamics lead to numerical instabilities when extending their moment sets accordingly. We solve this problem by training a recurrent neural network (RNN) that adjusts the QBMM quadrature to evaluate unclosed moments with higher accuracy. The proposed method is tested on a simple model of bubbles oscillating in response to a temporally fluctuating pressure field. The approach decreases model-form error by a factor of 10 when compared with traditional QBMMs. It is both numerically stable and computationally efficient since it does not expand the baseline moment set. Additional quadrature points are also assessed, optimally placed and weighted according to an additional RNN. These points further decrease the error at low cost since the moment set is again unchanged. This article is part of the theme issue 'Data-driven prediction in dynamical systems'.

Data Fusion With Latent Map Gaussian Processes

Abstract Multi-fidelity modeling and calibration are data fusion tasks that ubiquitously arise in engineering design. However, there is currently a lack of general techniques that can jointly fuse multiple data sets with varying fidelity levels while also estimating calibration parameters. To address this gap, we introduce a novel approach that, using latent-map Gaussian processes (LMGPs), converts data fusion into a latent space learning problem where the relations among different data sources are automatically learned. This conversion endows our approach with some attractive advantages such as increased accuracy and reduced overall costs compared to existing techniques that need to take a combinatorial approach to fuse multiple datasets. Additionally, we have the flexibility to jointly fuse any number of data sources and the ability to visualize correlations between data sources. This visualization allows an analyst to detect model form errors or determine the optimum strategy for high-fidelity emulation by fitting LMGP only to the sufficiently correlated data sources. We also develop a new kernel that enables LMGPs to not only build a probabilistic multi-fidelity surrogate but also estimate calibration parameters with quite a high accuracy and consistency. The implementation and use of our approach are considerably simpler and less prone to numerical issues compared to alternate methods. Through analytical examples, we demonstrate the benefits of learning an interpretable latent space and fusing multiple (in particular more than two) sources of data.

Physics-integrated hybrid framework for model form error identification in nonlinear dynamical systems

For real-life nonlinear systems, the exact form of nonlinearity is often not known and the known governing equations are often based on certain assumptions and approximations. Such representation introduce model-form error into the system. In this paper, we propose a novel gray-box modeling approach that not only identifies the model-form error but also utilizes it to improve the predictive capability of the known but approximate governing equation. The primary idea is to treat the unknown model-form error as a residual force and estimate it using dual Bayesian filter based joint input-state estimation algorithms. For improving the predictive capability of the underlying physics, we first use machine learning algorithm to learn a mapping between the estimated state and the input (model-form error) and then introduce it into the governing equation as an additional term. This helps in improving the predictive capability of the governing physics and allows the model to generalize to unseen environment. Although in theory, any machine learning algorithm can be used within the proposed framework, we use Gaussian process in this work. To test the performance of proposed framework, case studies discussing four different dynamical systems are discussed; results for which indicate that the framework is applicable to a wide variety of systems and can produce reliable estimates of original system’s states. Apart from this, the algorithm has also been tested for a case where the data has been taken from an experimental setup (Silver box dataset). The results produced further showcase the efficacy of the proposed framework.

A data assimilation model for wall pressure-driven mean flow reconstruction

This study establishes a continuous adjoint data assimilation model (CADA) for the reproduction of global turbulent mean flow from a limited number of wall pressure measurements. The model-form error induced by the Boussinesq assumption is corrected by a body force vector, which reinforces the eddy viscosity-based Reynolds force vector. The Stokes–Helmholtz decomposition is applied to this Reynolds force vector to isolate the crucial information contained with the Reynolds stress, and the primary-adjoint system is solved only for the anisotropic components. The CADA model is theoretically derived to minimize discrepancies between the wall pressure measurements and the numerical predictions of the primary-adjoint system. This minimization reveals the optimal anisotropic contribution of the Reynolds force vector. Four test cases are used for the assessment and validation of our CADA model. First, simulation of the wake in a flow over a cylinder demonstrates the ability of our CADA model to accurately recover the global fields from different regions of local synthetic wall measurements. Second, simulation of the flow over a backward-facing step illustrates that our CADA model can reconstruct a detached flow with a high Reynolds number. Third, simulation of the flow in a converging–diverging channel shows that our CADA model can reconstruct a strong adverse pressure-gradient flow. Fourth, simulation of the periodic hill flow further showcases the ability of our CADA model to predict complex flows. The method demonstrated here opens up possibilities for assimilating realistic observations, serving as a complement to our anisotropic DA scheme for future DA work.

Variance-based sensitivity analysis of dynamic systems with both input and model uncertainty

This paper develops a methodology to compute variance-based sensitivity indices for dynamic systems with time series inputs and outputs, while accounting for both aleatory and epistemic uncertainty sources, and both random process and random variable inputs. We present semi-analytical methods for computing sensitivity indices for linear systems with Gaussian random process inputs, and for the general case of nonlinear systems with non-Gaussian random process inputs. The novel elements in this approach are the treatment of model form error, quantifying the cumulative effects of uncertainty sources over time, and evaluating sensitivity indices for multi-physics models. Bayesian state and parameter estimation methods are incorporated to quantify the model uncertainty arising from unknown model parameters and model form errors, and sensitivity indices are computed before and after model updating. The proposed methods are illustrated for (a) a linear Timoshenko beam erroneously modeled as an Euler-Bernoulli beam, and (b) hypersonic flow behavior of a flexible panel represented by a coupled multi-physics nonlinear model.