A new artificial-intelligence architecture designed to handle the jagged, unruly mathematics of real-world systems has outperformed both conventional neural networks and established optimization software in a demanding test involving a French hydraulic valley. Called P1-KAN, the model is a new form of Kolmogorov-Arnold network, or KAN, developed by Xavier Warin of Électricité de France and EDF Lab Paris-Saclay. The research, published in Neural Computing and Applications, reports that P1-KAN can approximate irregular functions in many dimensions more accurately and with faster convergence than multilayer perceptrons, the workhorse architecture behind much of modern deep learning. In the hydraulic application, the system also produced better optimization results than competing KAN designs and classical deterministic methods based on dynamic programming. The finding arrives as researchers increasingly search for neural networks that are not only powerful, but also better suited to the complex, discontinuous and noisy calculations that govern energy systems, climate models, engineering design and financial decision-making.
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.
Subject of Research: Piecewise-linear Kolmogorov-Arnold networks for high-dimensional function approximation and hydraulic valley optimization
Subject of Research: Technology and Engineering
Article Title: P1-KAN: an effective kolmogorov-arnold network with application to hydraulic valley optimization
Article References: Warin, X. (2026). P1-KAN: an effective kolmogorov-arnold network with application to hydraulic valley optimization. Neural Computing and Applications, 38(16), Article 703. https://doi.org/10.1007/s00521-026-12354-y
Image Credits: AI Generated
DOI: 10.1007/s00521-026-12354-y
Keywords: P1-KAN, Kolmogorov-Arnold networks, deep learning, function approximation, hydraulic optimization, stochastic optimization, reservoir management, dynamic programming
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Everett Foxley. (August 28, 2026). P1-KAN: An Effective Kolmogorov-Arnold Network for Hydraulic Valley Optimization. Scienmag. https://scienmag.com/p1-kan-an-effective-kolmogorov-arnold-network-for-hydraulic-valley-optimization/
Everett Foxley. “P1-KAN: An Effective Kolmogorov-Arnold Network for Hydraulic Valley Optimization.” Scienmag, 28 August 2026, https://scienmag.com/p1-kan-an-effective-kolmogorov-arnold-network-for-hydraulic-valley-optimization/. Accessed 28 August 2026.
Everett Foxley. “P1-KAN: An Effective Kolmogorov-Arnold Network for Hydraulic Valley Optimization.” Scienmag. August 28, 2026. https://scienmag.com/p1-kan-an-effective-kolmogorov-arnold-network-for-hydraulic-valley-optimization/
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