Steel remains the backbone of modern industry, and the dominant way to make it is continuous casting, a process in which molten steel is poured into a water-cooled copper mold and solidifies into a thin, continuously withdrawn slab. The most dangerous and least observable part of that machinery is the interface where liquid steel meets the mold. No sensor can survive inside molten steel, so operators cannot directly measure the heat flux crossing that boundary. Yet knowing this flux in real time is what allows them to monitor the mold’s thermal behavior and avoid the catastrophic failures that force an emergency shutdown of the caster. A new open-access study published on 10 March 2026 in Computational Science and Engineering presents a Bayesian data assimilation framework that infers this invisible boundary heat flux from temperature measurements taken just inside the mold wall.
The research, carried out by Kabir Bakhshaei, Umberto Emil Morelli, Giovanni Stabile and Gianluigi Rozza, formulates the task as a stochastic inverse heat transfer problem. The transient heat flux at the mold-slab interface is treated as an unknown Neumann boundary condition in an unsteady three-dimensional heat conduction model of the mold. Instead of computing a single deterministic value for this flux, the team estimates its full probability distribution, updating the estimate sequentially as thermocouple readings arrive. This is a crucial distinction: a deterministic inversion returns one number, while the Bayesian approach returns a distribution with a mean and a variance, giving operators a quantified sense of how uncertain the estimate is at any moment.
The engine of the method is an ensemble-based filtering technique known as Ensemble-based Simultaneous Input and State Filtering with direct feed-through, or EnSISF-wDF. It is an extension of the Monte Carlo version of the Kalman Filter, a family of data assimilation methods originally developed to merge noisy observations with dynamic models. As its name implies, the filter estimates two things at once: the state of the system, here the temperature distribution throughout the mold, and the unknown input, here the unsteady heat flux at the hot face. The ensemble consists of many parallel model replicas, each carrying its own guess of the temperatures and the flux parameters. When a new set of thermocouple measurements becomes available, an ensemble Kalman gain blends the predicted observations with the actual ones, nudging every ensemble member toward a better joint estimate. The procedure is derivative-free and avoids propagating full covariance matrices, which makes it well suited to high-dimensional, highly nonlinear systems.
To keep the computational burden manageable, the authors parameterize the unknown heat flux using radial basis functions, or RBFs. Rather than estimating the flux at every boundary face of the mold, the method estimates the weights of a small number of basis functions whose spatial shape is fixed in advance. In the reported case study, only five RBFs were used, with centers defined by projecting five selected measurement points onto the hot surface of the mold. Since the setup uses 100 thermocouples, the ratio of measurements to unknown RBF weights is 20 to 1, a substantial reduction in the size of the inverse problem. The study compares two kernel types, the Gaussian kernel and the Multiquadric kernel, each governed by a shape parameter that controls how steeply and how far the basis function spreads across the domain.
The forward model itself is an unsteady three-dimensional heat conduction problem solved with OpenFOAM, an open-source computational fluid dynamics package that discretizes the domain into small control volumes. The mold is treated as a homogeneous, isotropic solid with constant material properties; surfaces exposed to air are assumed adiabatic, radiation is neglected, and the cold side in contact with cooling water at 350 K carries a convective boundary condition. For uncertainty quantification, the researchers coupled the MUQ library developed at MIT with ITHACA-FV, an in-house C++ library built on OpenFOAM by SISSA Mathlab. Notably, the forward model for this case study is cheap enough that the full-order model could be used directly, without reduced-order surrogates.
Because no physical experiment was available, the team generated synthetic measurement data by running a direct simulation with a known, time-varying true heat flux, a prescribed analytical function combining a quadratic spatial profile, a sinusoidal component with a maximum frequency of 0.1 Hz, and an exponentially decaying term over a 20-second horizon. The thermocouples sit at the plane y = 0.02 meters, a few millimeters inward from the hot boundary. The accuracy of each reconstruction was scored with a spatiotemporal relative error metric that compares the estimated and true heat fluxes across both space and time.
A major contribution of the paper is a systematic sensitivity analysis of hyperparameters that the authors say are not documented in the existing literature. These include the number of ensemble members, the RBF shape parameter, the time step, the observation span, and the shifting and scaling of the prior distribution on the RBF weights. The ensemble size showed a nuanced relationship with accuracy: increasing the number of samples initially reduced the spatiotemporal error, but beyond an optimum the error escalated sharply because the condition number of the linear system solved for the Kalman gain rises rapidly. With a Gaussian kernel the optimum was 375 ensembles, while the Multiquadric kernel needed only 300, implying lower computational cost for the same task.
The other hyperparameters told a consistent story. The optimal shape parameter was 0.5 for the Gaussian kernel and 3 for the Multiquadric kernel; an inappropriate value causes excessive overlap between basis functions and undermines the stability of the interpolation. Shifting the mean of the prior weights away from an exact projection of the true flux helped the Multiquadric case, reaching a best error of 6.3 percent at 30 percent shifting, whereas the Gaussian kernel performed best with no shifting at all, at 7.7 percent. Scaling the prior covariance by a factor kappa produced a non-monotonic error curve, with the Multiquadric optimum of 6.31 percent at kappa equal to 20 percent. Shorter observation spans were better for both kernels, with the minimum error at 0.40 seconds, and the optimal time step was 0.2 seconds for Multiquadric versus 0.1 seconds for Gaussian.
With all hyperparameters tuned, the Multiquadric version of the method emerged as the winner. Its final spatiotemporal relative error was 6.31 percent, compared with 7.59 percent for the Gaussian kernel, and its predicted heat flux contours at the hot boundary visually resembled the true flux more closely at time steps of 5, 10, 15 and 20 seconds. The predicted mean temperature at a probe point used by one of the thermocouples stayed within the 5 to 95 percent confidence envelope, as expected, since the thermocouples are the only locations where information enters the filter. Each update phase squeezed the confidence intervals further as additional measurements arrived, and the method also reconstructed the total heat flux integrated over the entire hot boundary with quantified uncertainty.
The authors are careful about scope. The study relied on a single synthetic true heat flux scenario, so it remains an open question whether the Multiquadric kernel would outperform the Gaussian, or whether some other basis function would beat both, under different operating dynamics. The forward model was inexpensive enough to run at full order, and the authors point to intrusive reduced-order modeling or surrogate models as the natural next step for real industrial problems, where the computational cost of the forward simulation becomes the bottleneck. They also flag the number of RBFs, currently fixed at five, as a hyperparameter that should be optimized according to the complexity of the flux. Even with those caveats, the demonstration that a probabilistic boundary condition can be estimated in real time from noisy thermocouple data, while explicitly accounting for both measurement noise and model error, marks a meaningful step toward monitoring systems that could warn continuous casting operators before thermal instabilities force a costly caster shutdown.
Subject of Research: Bayesian data assimilation for real-time estimation of the heat flux at the mold-slab interface in continuous casting
Article Title: Optimized Bayesian framework for inverse heat transfer problems using reduced order methods
Article References: Bakhshaei, K., Morelli, U. E., Stabile, G., & Rozza, G. (2026). Optimized Bayesian framework for inverse heat transfer problems using reduced order methods. Computational Science and Engineering, 3(1), Article 1. https://doi.org/10.1007/s44207-026-00009-8
Image Credits: AI Generated
DOI: 10.1007/s44207-026-00009-8
Keywords: continuous casting, inverse heat transfer problem, Bayesian inference, data assimilation, EnSISF, radial basis functions, heat flux estimation, ensemble Kalman filter, uncertainty quantification, OpenFOAM, steel manufacturing, real-time monitoring
News Source: Denise Maddox. (October 29, 2026). Bayesian filtering tracks heat flux in continuous casting molds in real time. Scienmag.
