The challenge the researchers set out to solve is fundamentally one of missing information. In a tunnel equipped with thermocouples, temperature sensors are typically placed at limited positions along the ceiling and walls because instrumenting every meter of a kilometer-long passage is impractical and expensive. Those sparse readings leave vast stretches of the temperature field unknown. Traditional approaches to filling in the gaps fall into two camps: purely data-driven machine learning models, which can interpolate between sensor points but often violate physical laws, and full computational fluid dynamics simulations, which respect physics but are far too slow to run during an actual emergency. The research team’s answer was to build a model that is both fast and physically faithful, learning from few measurements while remaining constrained by the governing equations of heat transport.

The centerpiece of the study is a Multi-source Physics-informed Kolmogorov–Arnold Network, abbreviated PIKAN. The Kolmogorov–Arnold network, or KAN, is a relatively recent alternative to conventional multilayer perceptrons. Instead of stacking layers of fixed activation functions applied to weighted sums, a KAN places learnable activation functions on the edges of the network, parameterized as one-dimensional functions. This design, rooted in the Kolmogorov–Arnold representation theorem, gives the architecture a different and often more efficient way of approximating complex multivariate functions. In this application, the authors strengthened their KAN by incorporating radial basis function layers, which enhance the network’s nonlinear approximation capacity and, crucially, ensure stable automatic differentiation, the mathematical machinery that allows a neural network to compute its own derivatives with respect to its inputs.

Why do derivatives matter so much here? The answer lies in the physics-informed part of the framework. The researchers embedded the tunnel ceiling heat diffusion equation directly into the loss function of the network, forcing the model’s predictions to satisfy the same partial differential equation that governs how heat spreads along the tunnel ceiling during a fire. Because the network must be differentiated to evaluate that equation, stable automatic differentiation is not a luxury but a necessity. Any numerical noise in the derivatives would corrupt the physics residual and destabilize training. By combining radial basis function layers with the KAN structure, the team obtained smooth, differentiable representations of the temperature field that could be checked against the diffusion equation at thousands of collocation points throughout the tunnel, even where no sensor exists.

The architecture itself is a coordinated ensemble rather than a single monolithic network. It comprises one temperature field network and multiple heat release rate networks, each dedicated to a candidate fire source, and all of them are jointly trained. The temperature field network learns the spatiotemporal distribution of temperature rise throughout the tunnel, while the heat release rate networks infer the intensity and location characteristics of each individual burning source. Training proceeds against two kinds of constraints simultaneously: the observed temperatures from sparse thermocouples, which anchor the solution to reality, and the embedded heat diffusion physics, which keeps the solution plausible everywhere else. The result is a joint inversion in which fire source locations, heat release characteristics, and the complete temperature rise field are all estimated together, each informing the others through the shared physical model.

Validation came from two demanding test cases. The first was a full-scale double-source tunnel fire experiment, the kind of physical trial that reproduces realistic fire behavior at true dimensions rather than in miniature. The second was a three-source tunnel fire scenario simulated with Fire Dynamics Simulator, the widely used computational tool for fire-driven fluid flow. These two cases complement each other: the full-scale experiment demonstrates that the method works against messy, real-world data, while the simulated three-source case probes whether the framework scales to more complex multi-source configurations with known ground truth for rigorous error assessment. Few studies in the fire safety literature attempt this kind of multi-source inversion with both experimental and numerical verification at these levels of complexity.

The reported performance is striking. In the double-source experimental case, the localization errors for the two fire sources were reduced to 0.89 meters and 1.06 meters respectively, meaning the algorithm could pinpoint each blaze within roughly a meter of its true position. In the more challenging three-source simulated scenario, all localization errors were controlled within 1.07 meters, with the minimum error reaching just 0.02 meters, effectively exact localization for one of the three sources. Beyond position, the reconstructed temperature fields agreed well with both the experimental measurements and the numerical simulation results, indicating that the model does not merely find the fires but correctly maps the thermal landscape they create. In direct comparison with the conventional Multi-source Physics-Informed Neural Network, or PINN, the PIKAN approach achieved higher localization accuracy and more reliable temperature reconstruction across the board.

The implications for tunnel safety are substantial. Knowing where a fire is burning, and how many fires are burning, determines where to direct ventilation to avoid pushing smoke toward evacuees, which exits remain tenable, and where firefighters should stage their approach. A temperature field reconstruction that is physically consistent, rather than a statistical guess, gives engineers confidence in the intermediate values between sensors, which matter for assessing structural damage to tunnel linings and for predicting when critical temperatures might be reached at specific locations. Because the method works from sparse thermocouple data already available in many modern tunnels, it could potentially be integrated into existing monitoring infrastructure, turning arrays of temperature sensors into a distributed fire intelligence system. The work also builds on a growing body of research by the same group and others applying intelligent algorithms, from support vector machine fusion for tunnel fire warning to BP neural network-based prediction of ceiling temperatures verified in full-scale experiments.

More broadly, the study signals a shift in how physics-informed machine learning is being engineered. Conventional PINNs, built on standard multilayer perceptrons, have sometimes struggled with spectral bias, an tendency to learn low-frequency components of a solution first and converge slowly on sharp or multi-scale features, exactly the kind of features that characterize the steep thermal gradients near a fire source. Kolmogorov–Arnold networks, with their learnable edge activation functions and enhanced approximation capacity through radial basis functions, represent a promising remedy, and this study is among the first to demonstrate their advantage in a real safety-critical inverse problem involving multiple simultaneous sources. As the authors’ results show, a neural network that respects the heat diffusion equation and possesses the right approximation machinery can turn a handful of ceiling temperature readings into a complete, accurate, physically consistent picture of a multi-source tunnel fire, a capability that could one day mean the difference between a managed evacuation and a catastrophe.

Subject of Research: Physics-informed neural networks for fire source localization and temperature field reconstruction in multi-source tunnel fires

Article Title: Physics-informed Kolmogorov–Arnold network for fire source localization and temperature field reconstruction in multi-source tunnel fires

Article References: Li, Y., & Sun, B. (2026). Physics-informed Kolmogorov–Arnold network for fire source localization and temperature field reconstruction in multi-source tunnel fires. Applied Intelligence, 56(15), Article 441. https://doi.org/10.1007/s10489-026-07504-7

Image Credits: AI Generated

DOI: 10.1007/s10489-026-07504-7

Keywords: tunnel fire, fire source localization, temperature field reconstruction, Kolmogorov–Arnold network, physics-informed neural network, PIKAN, heat diffusion equation, inverse problem, Fire Dynamics Simulator, machine learning, fire safety, sparse sensor data