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

At the heart of the development is a question that has occupied mathematicians and computer scientists for decades: how can a machine approximate a complicated function of many variables? A conventional multilayer perceptron, or MLP, processes information through layers of artificial neurons. Each neuron applies a weighted sum to its inputs and then passes the result through a nonlinear activation function such as a rectified linear unit. The network learns the weights connecting neurons, gradually bending a high-dimensional input-output relationship into a useful approximation. KANs take a different route. Inspired by the Kolmogorov-Arnold representation theorem, they place learnable one-dimensional functions on connections between nodes rather than relying primarily on fixed activation functions inside neurons. In principle, this lets the model build a complex multivariable function by combining simpler functions of individual coordinates. The approach has generated intense interest because the learned functions can sometimes be inspected directly, offering a possible advantage in interpretability over opaque deep networks.

The mathematical theorem behind KANs does not automatically guarantee that every practical implementation will work well. The original Kolmogorov-Arnold representation concerns continuous functions and provides an existence result, but translating that insight into a stable, efficient learning algorithm is a separate engineering and analytical challenge. Real optimization problems often involve functions with sharp bends, kinks, regime changes or other forms of irregularity. Such behavior can arise when a system switches between operating constraints, when a physical process changes abruptly, or when a small variation in an input produces a disproportionately large change in the output. Smooth spline-based KANs can be highly effective when the target function is itself smooth, but their performance may degrade when the function contains irregular structure. Warin’s P1-KAN is designed around this difficulty, using piecewise-linear components intended to represent abrupt changes without requiring the network to force them into an overly smooth shape.

The “P1” designation refers to first-order, piecewise-polynomial behavior. Rather than representing each learnable connection with a globally smooth curve, the architecture constructs functions from local linear segments. A piecewise-linear function changes slope at selected breakpoints, allowing it to approximate a sudden transition while retaining a comparatively simple computational form. This design can also make optimization more manageable: the model does not need to adjust a large collection of highly flexible smooth basis functions to reproduce a sharp feature. P1-KAN therefore occupies a different point in the trade-off between flexibility, accuracy and computational cost. The study presents universal approximation theorems for several versions of the architecture, meaning that, under specified conditions and with sufficient model capacity, the networks can approximate broad classes of target functions to arbitrary precision. The authors also derive error estimates for cases in which the underlying Kolmogorov-Arnold representation functions possess regularity, providing a theoretical framework for understanding when the model should perform well.

To test the practical consequences of the theory, the researchers first examined simple regression problems in which the networks had to learn known mathematical relationships. These controlled experiments make it possible to separate the architecture’s behavior from the complications of a large industrial data set. According to the study, P1-KAN achieved higher accuracy than MLPs and reached useful solutions more quickly during training. In machine learning, convergence speed measures how rapidly an optimization procedure reduces its error or objective function. Faster convergence can lower computational expense and can be crucial when a model must be retrained repeatedly as conditions change. The results also showed that P1-KAN was especially effective when the target functions were irregular. For smooth functions, its accuracy was similar to that of the original spline-based KAN, suggesting that the piecewise-linear design does not sacrifice performance simply because it is built to handle rougher mathematical terrain. The comparisons included several other KAN variants, reflecting the rapidly expanding ecosystem of architectures based on the same broad idea.

The decisive demonstration involved the optimization of a hydraulic valley in France, an industrial problem connected to the operation of water reservoirs and hydropower infrastructure. Reservoir optimization requires decisions about how water should be stored and released over time. Operators must balance competing objectives, such as electricity production, water availability, downstream constraints and the uncertain arrival of future inflows. A release that maximizes power generation today may reduce flexibility tomorrow; retaining water may preserve future options but miss a valuable opportunity in the present. Mathematically, the problem can be framed as a sequential control task in which the optimal action depends on the current state of the hydraulic system and on uncertain future conditions. The value function—the estimated long-term benefit associated with a particular state—can become highly nonlinear and irregular, especially when operational limits or discrete decisions are involved. That makes it an exacting environment for a function-approximating neural network.

Practitioners have traditionally approached such problems with dynamic programming, a method formalized in influential work on sequential decision-making and stochastic control. Dynamic programming breaks a complex multistage problem into linked subproblems and uses a recursive relationship to calculate the value of decisions over time. In a simple setting, the method can be extraordinarily powerful. But its computational demands grow rapidly as the number of state variables increases, a difficulty commonly described as the curse of dimensionality. Fine-grained representations of reservoir levels, inflows, market conditions and other variables can require enormous memory and processing time. Neural networks offer a way to approximate the value function or policy without explicitly enumerating every possible state. They can generalize from sampled scenarios, potentially making optimization feasible in settings where a grid-based dynamic-programming calculation becomes unwieldy. Yet that advantage depends on the network learning the system’s irregularities rather than smoothing them away or converging to a misleading solution.

In the hydraulic valley experiment, P1-KAN optimized the system more effectively than the other KAN networks tested and also surpassed the classical deterministic tools used by practitioners, the study reports. The result does not mean that a neural network has replaced hydraulic expertise or eliminated uncertainty from reservoir management. Instead, it indicates that the architecture’s particular way of representing nonlinear functions may be valuable for a difficult class of industrial optimization problems. A model can perform well in this setting because it captures the structure of the objective function and constraints with fewer approximation errors, because it trains more efficiently, or because its local piecewise representation adapts better to changes in operating regimes. The article’s findings support the idea that architecture matters: simply applying a larger or deeper network is not always the best answer. Matching the mathematical properties of a model to the structure of the problem may deliver larger gains than adding layers or parameters.

The implications extend well beyond water management. Kolmogorov-Arnold networks are being explored for mechanics, medical-image analysis, time-series forecasting, survival analysis, physics-informed neural networks and other applications in which interpretability or nonlinear approximation is important. P1-KAN could be relevant wherever a system contains thresholds, discontinuities or sharply changing responses, from energy dispatch and infrastructure planning to stochastic control and engineering design. At the same time, the study highlights why enthusiasm should be paired with careful validation. The experiments described in the article establish theoretical approximation properties and report strong results on selected regression and hydraulic optimization tasks, but they do not prove that P1-KAN will outperform every architecture on every data set. Other research has reported limitations for KANs on noisy functions, and comparisons between neural-network families can depend heavily on training procedures, parameter counts, data quality and hardware implementation. The authors state that data are available from them on request, which could allow independent researchers to examine the benchmarks and reproduce the findings.

The broader significance of P1-KAN is that it brings a more problem-aware philosophy to the current neural-network race. Artificial intelligence has often advanced by scaling: more data, more parameters and more computing power. But many scientific and industrial systems are governed by equations, constraints and abrupt physical or economic transitions that generic architectures may represent inefficiently. A network built from adaptable one-dimensional functions can expose a different set of mathematical building blocks, while a piecewise-linear version can target irregularity directly. Warin’s results suggest that this combination can make a measurable difference, both in clean mathematical tests and in a real optimization problem with practical consequences. If future studies confirm the gains across additional reservoirs, uncertainty models and high-dimensional control tasks, P1-KAN may become part of a new generation of scientific machine-learning tools—systems designed not merely to fit data, but to respect the shape of the problems they are asked to solve.