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Causal Local States: Scalable Simultaneous Causal Network Inference and Forecasting for Dynamical Systems

This paper introduces Causal Local States (CLS), a scalable framework that simultaneously infers heterogeneous causal interaction networks and forecasts dynamical systems by independently selecting optimal predictive neighborhoods for each node, thereby achieving high-fidelity network reconstruction and prediction accuracy comparable to models with known ground-truth structures.

Original authors: Jonas Braun, Fabian Fischbach, Daniel Köglmayr, Sebastian Baur, Christoph Räth

Published 2026-10-01

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CC BY 4.0

Original authors: Jonas Braun, Fabian Fischbach, Daniel Köglmayr, Sebastian Baur, Christoph Räth

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The world is full of systems that move and change in complex ways: the swirling currents of the atmosphere, the flow of electricity through a national grid, or the shifting populations of an ecosystem. To understand these systems, scientists often rely on machine learning, which can predict future behavior with remarkable accuracy. However, these powerful tools usually act as black boxes. They take in raw data and spit out a forecast, but they do not reveal why they made that prediction or which specific parts of the system are actually influencing each other. Without this insight, we cannot truly understand the mechanics driving the change, nor can we easily trust the predictions when conditions shift. To solve this, researchers need a way to not only forecast the future but also map the invisible web of cause and effect that connects the parts of the system together.

A new approach called Causal Local States, or CLS, offers a solution by combining the ability to predict with the ability to explain. Traditional methods for finding these connections often rely on a single, rigid rule applied to the entire system, such as setting a universal threshold for what counts as a connection. This works poorly for complex systems where different parts behave differently; a rule that fits one section might fail completely in another. The researchers behind CLS, led by Jonas Braun and colleagues, developed a method that treats each part of a system individually. Instead of forcing the whole system into one mold, the method looks at every single variable—like a specific weather station or a power grid node—and asks a simple question: which other variables are actually needed to predict this one?

The process works like a careful filter. First, the computer scans all possible connections to a specific point and ranks them based on how much they help predict the future of that point. It then tests these candidates, starting with the most promising ones, to see if adding more information actually improves the forecast. If a variable does not add value, it is discarded. This happens independently for every single point in the system. Once the best, smallest group of neighbors is found for each point, the method stitches these local groups together to form a complete map of the system's interactions. This map is then used to run the forecast, ensuring that the prediction is built only on the relationships that truly matter.

The team tested this framework on three different types of systems, each more difficult than the last. The first test involved a collection of independent pairs of mathematical models that mimic chaotic weather patterns. In this setup, the two models in a pair interact with each other, but they have no connection to any other pair. A standard prediction model that looked at all the data at once failed to make accurate forecasts because it was overwhelmed by the sheer amount of information. The CLS method, however, successfully identified the correct pairs and separated them from the rest. It reconstructed the hidden network with high accuracy and produced forecasts that were just as good as a model that was already given the correct map of connections.

In the second test, the researchers applied the method to a system known as Lorenz96, which simulates a chain of interacting variables often used to study atmospheric dynamics. Here, the goal was to see if the method could find the specific pattern of connections that governs the system. The CLS framework not only recovered the correct structure but also found that including a few extra, nearby variables actually improved the prediction, even though those variables were not part of the strict mathematical definition of the system. This suggests that for the purpose of forecasting, the most useful connections are not always the most obvious ones. The method produced forecasts that matched the performance of a model supplied with the true, underlying equations of the system.

The final and most challenging test involved a realistic simulation of the United Kingdom's high-voltage power grid. This is a complex network of 120 nodes where electricity flows and oscillates. Predicting the behavior of such a grid is notoriously difficult because standard prediction tools often become unstable and produce wild, incorrect results. The researchers adapted their method to use a specialized model better suited for this type of oscillating system. The result was striking: the CLS framework reconstructed the actual network topology of the power grid with near-perfect accuracy. It identified the true connections between the nodes and, using this inferred map, generated forecasts that were comparable to those made by a model that already knew the true structure of the grid.

The significance of this work lies in its ability to handle high-dimensional systems without needing prior knowledge of how they are connected. By inferring the network structure directly from the data and doing so in a way that is optimized for prediction, the method bridges the gap between understanding and forecasting. It proves that it is possible to take a massive, unstructured collection of data, figure out which parts are talking to each other, and use that understanding to predict the future of the whole system. While the method has so far been tested on simulations, the researchers note that it holds promise for real-world applications in fields like weather forecasting and power grid management, where knowing both the "what" and the "why" of a prediction is essential. The study demonstrates that by breaking a complex problem down into small, local pieces and solving each one independently, we can build a clear and accurate picture of the most intricate systems in our world.

Technical Summary: Causal Local States (CLS)

Problem Statement
Machine learning methods excel at forecasting complex, high-dimensional dynamical systems but often function as "black boxes," offering no insight into the underlying interaction networks driving the dynamics. Conversely, causal discovery methods can reconstruct interaction networks from observational data but typically do so without regard for whether the inferred structure supports accurate forecasting. Existing approaches that attempt to combine these tasks rely on a single global hyperparameter (e.g., a fixed causal threshold or neighborhood size). This assumption fails for heterogeneous systems where subsystems possess different dynamics and scales, making a global parameter incapable of recovering all local neighborhoods correctly. Furthermore, standard data-driven forecasting models, such as Reservoir Computing (RC) and its successor Next-Generation Reservoir Computing (NGRC), process all variables simultaneously. In high-dimensional settings, this leads to computational infeasibility and a lack of interpretability regarding the underlying interaction network.

Methodology
The authors introduce Causal Local States (CLS), a framework that simultaneously infers an approximate Granger-causal interaction network and forecasts system dynamics. The core concept is to decompose a high-dimensional system into $d$ independent local prediction problems, one for each "core node."

The CLS framework operates through a three-step procedure for each node independently:

Causal Filtering: A bivariate directed causal measure (e.g., Transfer Entropy or Convergent Cross Mapping) ranks all other variables as candidate neighbors. This step generates a nested sequence of candidate neighborhoods, reducing the search space from $O(2^d)$ to a manageable set of size $d_{max}$ .
Wrapper Selection: A black-box predictive model (specifically NGRC) is trained on each candidate neighborhood to forecast the core node. The model evaluates the predictive performance (e.g., Mean Squared Error) of these candidates. The algorithm selects the smallest neighborhood that achieves near-optimal performance based on a relative improvement criterion.
Backward Elimination: To remove spurious or redundant features that may have passed the filter or wrapper stages, the algorithm iteratively removes the least informative neighbor. A neighbor is eliminated if its removal does not significantly degrade prediction accuracy, adhering to the causal Markov condition which posits that a variable is conditionally independent of its non-descendants given its parents.

The inferred local neighborhoods are combined into a global neighborhood matrix. The full system is then forecast in parallel: each node is predicted using only its specific inferred neighborhood, and one-step predictions are recursively fed back to generate multi-step trajectories. The framework is model-agnostic regarding the causal measure and the wrapper model, though the authors utilize NGRC for its efficiency and ability to handle nonlinearities.

Key Contributions

Unified Framework: CLS bridges causal discovery and predictive modeling, treating variable selection as an operational form of Granger causality where variables are included only if they improve forecast accuracy.
Scalability: By inferring neighborhoods independently for each node, the method avoids the curse of dimensionality. The computational complexity scales linearly with the number of nodes when parallelized, making it feasible for high-dimensional systems where monolithic models fail.
Adaptability: Unlike previous methods using global thresholds, CLS determines the optimal neighborhood size and composition locally for each node, accommodating heterogeneous systems with varying dynamics.
Explainability: The output includes both the system forecast and the explicit interaction network (neighborhood matrix), providing insights into which variables drive the prediction for each node.

Results
The authors validated CLS on three benchmarks of increasing difficulty:

Composite Lorenz63–Rössler Systems (up to 90 dimensions): CLS successfully separated non-interacting attractor pairs, recovering the block-diagonal structure of the interaction network with high fidelity. It achieved a True Positive Rate (TPR) above 0.8 and suppressed False Positive Rates (FPR) to near zero after backward elimination. Crucially, CLS produced forecasts comparable to a model supplied with the true network, whereas a monolithic NGRC trained on the full state vector failed to predict the system.
Lorenz96 System (40 dimensions): CLS recovered the banded coupling structure of the system. The inferred neighborhoods contained slightly more nodes than the strict governing-equation parents (due to the inclusion of nearby variables that encode predictive information per Takens' embedding theorem), yet the resulting forecasts matched the performance of a model using the true adjacency matrix.
UK Power Grid (Higher-order Kuramoto Model, 120 nodes): Applied to a real-world network topology, CLS paired with a domain-informed wrapper (Kuramoto-NGRC) reconstructed the network topology almost perfectly (TPR ~96%, FPR ~1.9% after elimination). The forecast accuracy was on par with a model provided with the true network. The study highlighted the importance of matching the wrapper model to the system dynamics, as a standard polynomial NGRC wrapper failed on this oscillator network.

Significance and Claims
The paper claims that CLS represents a step toward explainable and scalable forecasting for complex dynamical systems. By explicitly inferring the interaction structure rather than leaving it implicit within a monolithic model, CLS provides a direct, inspectable output alongside the forecast. The authors emphasize that the framework does not require prior knowledge of the network and naturally scales to high-dimensional systems through its parallel, per-node inference architecture. While the method shows strong potential, the authors note that it has not yet been applied to real-world datasets and that hyperparameter optimization across the many local models remains an open challenge. The work suggests that selecting neighborhoods based on forecast skill, rather than strict causal graph recovery, can yield structures that are more effective for prediction, even if they differ from the nominal governing equations.

Technical Summary: Causal Local States (CLS)

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