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

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

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

Published 2026-08-19

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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 (http://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 around us is often governed by complex systems where countless parts interact in ways that are difficult to predict. Power grids, weather patterns, and ecosystems are all examples of high-dimensional dynamical systems, meaning they involve many variables changing over time in a non-linear fashion. For decades, scientists have relied on machine learning to forecast how these systems will evolve. While these models can be incredibly accurate, they often function as black boxes: they take raw data and spit out a prediction without revealing which specific parts of the system are actually driving the change. This lack of transparency makes it hard to understand the underlying causes or to trust the model when conditions change. At the same time, other methods exist that try to map out the connections between variables to understand the system's structure, but these approaches often ignore whether the resulting map is actually useful for making future predictions. The challenge has been to find a way to do both at once: to uncover the hidden network of interactions while simultaneously using that knowledge to forecast the system's future with high accuracy.

A team of researchers has developed a new framework called Causal Local States to solve this problem. Instead of treating the entire system as one massive, indivisible block of data, this approach breaks the problem down into smaller, manageable pieces. The core idea is that to predict the future behavior of any single part of a complex system, you do not need to know the state of every other part. You only need to know the state of the specific neighbors that directly influence it. The researchers designed a method that independently figures out these small groups of influential neighbors for every single variable in the system. It works by testing different combinations of variables to see which ones, when fed into a prediction model, yield the most accurate forecast for that specific part. If adding a variable does not improve the prediction, it is discarded as irrelevant. This process effectively filters out the noise and identifies the true causal drivers for each node.

Once the researchers have identified these local neighborhoods for every part of the system, they combine them into a complete map of the interaction network. This map is not just a theoretical reconstruction; it is immediately put to work. The system is then forecast by running a separate prediction model for each node, using only the data from its own small, identified neighborhood. Because each prediction depends only on a few nearby variables rather than the entire system, the method avoids the computational difficulties that usually arise when dealing with high-dimensional data. The researchers tested this approach on three different types of systems, each more difficult than the last. The first test involved a collection of independent mathematical systems that were known to have no connection to one another. The method successfully identified that these systems were separate, correctly ignoring the variables from one system when predicting the others, and it achieved this without confusing unrelated variables as causes.

The second test used a system known as the Lorenz96 model, which simulates atmospheric dynamics and features a specific pattern of connections where each part interacts primarily with its immediate neighbors. In this scenario, the researchers found that the method could recover the banded structure of the connections with high fidelity. Interestingly, the method sometimes kept a few extra connections that were not part of the original mathematical definition of the system. However, these extra connections did not hurt the prediction; in fact, they helped the model perform just as well as a model that was given the perfect, true network from the start. This suggests that for the purpose of forecasting, including variables that are closely related, even if they are not direct causes, can be beneficial. The final and most challenging test involved a dynamical model of the United Kingdom's power grid, a real-world network of 120 nodes. Here, the researchers paired their method with a specialized prediction model designed for oscillating systems. The result was a reconstruction of the power grid's network that was almost perfect, matching the true topology of the grid. The forecasts generated by this reconstructed network were just as accurate as those generated by a model that already knew the true network structure.

The significance of this work lies in its ability to make complex forecasting both scalable and explainable. By inferring the interaction network directly from the data and using that network to drive the predictions, the method provides a clear view of which variables matter for the future of the system. It avoids the need for a single, global rule to decide which connections are important, recognizing instead that different parts of a system may have different needs. This allows the method to handle systems where the rules of interaction vary from one part to another. The researchers demonstrated that this approach works even when the underlying network is completely unknown, and it scales efficiently to systems with many variables. While the current study focused on simulated data and mathematical models, the framework is designed to be adaptable to real-world applications. The ability to simultaneously uncover the structure of a complex system and predict its future evolution represents a step forward in making machine learning more transparent and reliable for understanding the intricate dynamics of the world around us.

Technical Summary: Causal Local States (CLS)

Problem Statement

Machine learning methods, particularly deep learning and reservoir computing, have demonstrated remarkable accuracy in forecasting complex, high-dimensional, nonlinear dynamical systems (e.g., power grids, atmospheric models). However, these models typically function as "black boxes," offering no insight into the specific interactions driving the system's dynamics. Conversely, causal discovery methods can reconstruct interaction networks from observational data but often ignore whether the inferred structure actually supports accurate prediction.

Existing approaches that attempt to combine network inference with forecasting rely on a single global hyperparameter—such as a fixed causal threshold or a uniform neighborhood size—to determine the interaction structure for the entire system. This approach fails for heterogeneous systems where different subsystems possess distinct dynamics and scales; a global threshold cannot simultaneously recover the correct structure for all nodes. Furthermore, standard data-driven forecasting methods (like Next-Generation Reservoir Computing, NGRC) process all variables simultaneously, leading to the "curse of dimensionality" where computational costs explode and model interpretability vanishes in high-dimensional settings.

Methodology: Causal Local States (CLS)

The authors introduce Causal Local States (CLS), a unified framework that simultaneously infers an approximate Granger-causal interaction network and forecasts system dynamics. The core philosophy is that for each node in a system, only a specific, small subset of other variables (its "local neighborhood") is necessary to predict its future state near-optimally.

The CLS framework operates through a three-step procedure performed independently and in parallel for each core node $X_i$ :

Causal Filtering:
A directed, bivariate causal measure (e.g., Transfer Entropy or Convergent Cross Mapping) is used to rank all other variables as candidate neighbors for the core node. This creates a nested sequence of candidate neighborhoods, ordered by causal strength. This step reduces the search space from an exponential $O(2^d)$ to a linear $O(d_{max})$ set of candidates, where $d_{max}$ is a hyperparameter bounding the neighborhood size.
Wrapper Selection:
A black-box predictive model (specifically NGRC in this work) acts as a "wrapper." It is trained and evaluated on the candidate neighborhoods generated in the filtering step. The wrapper selects the smallest neighborhood that achieves near-optimal predictive performance for the core node. The selection criterion is based on a relative improvement in prediction loss (e.g., Mean Squared Error) compared to the previous candidate size.
Backward Elimination:
The selected neighborhood may still contain redundant or spurious variables. A backward elimination step iteratively removes the least informative neighbor. A variable is removed only if its exclusion does not significantly degrade the prediction accuracy (within a defined threshold). This step refines the neighborhood to an approximate causal set, leveraging the causal Markov condition: given a variable's true parents, it is conditionally independent of other non-descendants.

Global Prediction:
Once the local neighborhoods are inferred for all $d$ nodes, they are combined into a global neighborhood matrix. The system is then forecast using a local-states approach: a separate predictive model is trained for each node using only its inferred local neighborhood. One-step predictions are concatenated and fed back recursively to generate multi-step trajectories for the full system. This architecture allows the framework to scale naturally to high-dimensional systems and enables parallel processing.

Key Contributions

Unified Framework: CLS bridges causal discovery and predictive modeling, treating the interaction network not as a separate inference task but as a means to optimize forecasting performance.
Local-Node Independence: Unlike previous methods using global hyperparameters, CLS infers the neighborhood for each node independently based on local criteria. This allows the framework to handle heterogeneous systems where interaction strengths and scales vary significantly across the network.
Wrapper-Based Causality: The paper operationalizes Granger causality as a variable selection problem. By using predictive performance as the scoring metric, the method selects variables that genuinely improve forecasts, effectively distinguishing true causal drivers from spurious correlations.
Scalability: By decomposing the global prediction problem into $d$ small, local problems, CLS avoids the computational infeasibility of training monolithic models on high-dimensional state vectors.

Results

The authors validated CLS on three benchmarks of increasing difficulty:

Composite Lorenz63–Rössler Systems (up to 90 dimensions):
- Task: Reconstructing a system composed of non-interacting attractor pairs.
- Outcome: CLS successfully separated the independent attractors, recovering the block-diagonal structure of the interaction network with high fidelity (True Positive Rate > 0.8, False Positive Rate near zero after elimination).
- Forecasting: CLS achieved valid prediction steps (VPS) comparable to a model supplied with the true network. In contrast, a monolithic NGRC trained on the full state vector failed to predict the system entirely due to memory constraints and the curse of dimensionality.
Lorenz96 System (40 dimensions):
- Task: Forecasting a chaotic system with a banded coupling structure.
- Outcome: CLS recovered the underlying coupling pattern. While the inferred neighborhoods contained slightly more nodes than the strict governing-equation parents (likely due to indirect predictive information allowed by Takens' embedding theorem), the resulting forecasts were on par with a model trained on the true adjacency matrix.
- Insight: The results suggest that the structure yielding the best predictions does not necessarily coincide with the strict governing-equation graph; retaining nearby, indirectly connected nodes can improve forecast skill.
UK Power Grid (Higher-Order Kuramoto Model, 120 nodes):
- Task: Forecasting a real-world network topology with higher-order interactions.
- Outcome: Standard polynomial NGRC failed on this oscillator network. By pairing CLS with domain-informed wrappers (Sine-NGRC and Kuramoto-NGRC), the framework reconstructed the network topology with high accuracy. Specifically, using Kuramoto-NGRC, the backward elimination step reduced the False Positive Rate from 10.62% to 1.87% while maintaining a True Positive Rate of 96.1%.
- Forecasting: The CLS framework using Kuramoto-NGRC achieved prediction accuracy comparable to a model given the true network. In contrast, the Sine-NGRC wrapper collapsed when using its own inferred adjacencies (dropping from 601 to 140 valid prediction steps), highlighting that model selection is critical for performance on this specific system.

Significance and Claims

The paper claims that CLS represents a step toward explainable and scalable forecasting of complex systems. By explicitly inferring the interaction network alongside the forecast, the method provides a direct, inspectable output: for every variable, it reports the specific set of other variables its prediction relies on.

The authors emphasize that CLS does not require prior knowledge of the underlying network. Its ability to infer structure directly from unstructured time series data, while simultaneously optimizing for prediction accuracy, addresses the limitations of both pure causal discovery (which ignores prediction utility) and pure black-box forecasting (which lacks interpretability). The framework is presented as a generalizable tool where the causal measure and the predictive wrapper are exchangeable components, adaptable to various system characteristics.

The paper concludes by noting that while CLS shows strong potential, it has not yet been applied to real-world datasets (e.g., financial time series or actual weather data), and challenges remain regarding hyperparameter optimization for the multiple local models involved.