Reconstruction of the underlying networks with high fidelity and forecasts on par with a model that is supplied with the true network are achieved, providing a step toward explainable and scalable forecasting of complex systems.
Abstract
Machine learning methods predict many real-world systems with remarkable accuracy, but they are typically treated as black boxes that offer no insight into which interactions drive the dynamics. Causal discovery methods reconstruct the interaction network from observational data, but without regard to whether the inferred structure supports prediction. Existing approaches combining both tasks rely on a single global hyperparameter, such as a causal threshold or a fixed neighborhood size, which cannot recover the structure of heterogeneous systems. Here we introduce causal local states (CLS), a framework that simultaneously infers an approximate Granger-causal interaction network and forecasts the system dynamics. For each node independently, we select the smallest set of neighbors that allows a predictive model to forecast the node near-optimally, and the resulting neighborhoods are then combined for a forecast of the full system. On three benchmarks of increasing difficulty, we achieve reconstruction of the underlying networks with high fidelity and forecasts on par with a model that is supplied with the true network, providing a step toward explainable and scalable forecasting of complex systems.
Many real-world systems can be modelled as complex networks whose collective behaviour is governed by hidden interactions between nodes. Existing methods for inferring these interactions typically require controlled perturbations, time-resolved observations or multiple independent snapshots, all of which are often unavailable in practice. Here we show that class-based coupling strengths can be inferred from a single snapshot of node states when the system is observed close to a relative equilibrium. In this regime, all nodes share a common velocity, which can be absorbed into an effective class bias, transforming the inverse problem into a homogeneous linear system. The coefficients of this linear system are determined entirely by the observed local neighbourhoods and their coupling mechanism, enabling the application to arbitrary known coupling functions. We validate the approach on three different linear and nonlinear dynamical systems, recovering relative class-based couplings and, in special cases, absolute couplings. These results show that spatial heterogeneity can substitute for temporal sampling, enabling single-snapshot inference of hidden coupling strengths in networked dynamical systems.
Moritz Lampert, Dominic Grün, Ingo Scholtes· 0 citations
Causal discovery aims to uncover the underlying causal relationships given data generated from a system. The goal, however, is not merely to predict causal edges given data, but also to be able to interpret and explain either observed or hypothesized phenomena, such as a particularly large causal effect. We consider this task of conditional causal discovery and cast it as a Bayesian inference problem, in which we target the posterior over causal graphs and parameters conditional on an event such as a causal-effect constraint. Unfortunately, this poses a computational challenge: existing approaches to Bayesian causal discovery struggle when the event has small posterior mass. To address this, we adapt rare-event estimation techniques to perform inference the joint graph-parameter space. Our method gradually drives a particle population toward the constrained region while maintaining samples that approximate the conditional posterior. Empirical evaluation on synthetic graphs validates the accuracy of our approach at small and large scales, and we show in a case study on the Sachs protein dataset how our method can be used to aid scientific exploration by providing pathway-level summaries.
Cixuan Zhang, Guy Van den Broeck, Benjie Wang· 0 citations
A calibrated stochastic world model can reveal how uncertain a future is without revealing why it branches. The same conditional future law can arise because an observation aliases physical states or because dynamics remain random after the declared full state is fixed. We prove that ordinary transitions cannot identify these two sources, even for a perfect probabilistic predictor. ClosurePairs makes them identifiable by crossing compatible microstates with repeated exogenous disturbances and estimating state, noise, and state-noise interaction variance. The central consequence is operational: under finite hierarchical sampling, forecast difficulty governs the useful compute scale, while the alias/process composition provides complementary information about its direction-resolving the current state or sampling future randomness. ClosurePairs recovers source attribution at unchanged likelihood, reduces equal-budget decomposition error in a nonlinear interaction benchmark, and supports observation-only routing. On exact-marginal MetaWorld twins, an output-only allocator is at chance while a Closure-supervised probe on frozen JEPA-WM features routes 89.8-100%. In an independent ManiSkill PushCube confirmation, a stochastic RSSM's outputs and latents remain at chance, whereas an RGB-only Closure probe routes 100% under both ID and geometry/camera OOD over five seeds, matching direct allocation rather than exceeding it. Across five unseen allocation menus, the same Closure probe routes 92.5%/90.4% ID/OOD with no new oracle labels, versus 37.9%/32.9% for a frozen direct allocator. ClosurePairs is therefore an identifiable, reusable mechanism target that cannot be recovered from forecast quality alone.
Mechanisms of interaction in spreading models are central to our quantitative understanding of networked contagion processes, from disease transmission to opinion dynamics. Yet, while empirical data can reveal who interacts with whom, they rarely provide direct information about how interactions drive spreading, leaving the underlying mechanism to be inferred from observed dynamics, and selected among competing hypotheses. We propose a general framework for model selection in binary-state spreading processes on networks and show that asymptotic approximations in the thermodynamic limit can accurately predict inference outcomes in finite systems. By systematically exploring a broad parameter space, we characterize the detectability of six archetypal spreading mechanisms commonly used in the literature and find that accuracy generally increases in sparse networks, which are prevalent in real-world systems, and near phase transitions, such as the epidemic threshold of simple contagion processes. We further assess the prevalence of these mechanisms across a diverse set of empirical datasets, highlighting the impact of data preprocessing on model recovery. Our results show that statistical model selection can fail under common conditions and suggest new directions for overcoming these limitations.
Javier Ureña-Carrión, Tiago P. Peixoto, G. Íñiguez· 0 citations
Non-stationary time series are common in many real-world domains, including infectious disease spread, where the underlying relationships between variables evolve over time. However, most existing forecasting methods assume stationarity and fail to capture changing causal dynamics. To address this challenge, we propose the Causal Regime Bayesian (CaReBayes) forecasting framework, which integrates regime detection, causal discovery, and Bayesian forecasting within a unified approach. CaReBayes segments time series into regimes using temporal causal discovery, fits a Bayesian structural autoregressive model for each regime, classifies the current regime, then performs regime-specific forecasting with uncertainty quantification. The framework introduces methodological advances: a grid-search procedure for automated regime-dependent causal discovery, a classification method that assigns future observations to regimes based on learned Bayesian structures, and regime-conditioned Bayesian structural forecasting. Across both synthetic and Ontario COVID-19 time series data, CaReBayes outperforms benchmark models for time series forecasting. In addition to improved forecasting performance, it produces regime-dependent causal graphs that summarize candidate structural relationships in the system, enhancing interpretability.
Understanding and controlling complex dynamical systems often requires executing thousands of numerical simulations across vast parametric landscapes, which is time-consuming. Machine learning surrogates significantly accelerate simulation by predicting state trajectories across different initializations and parameter values. However, surrogate models are specialized to one simulation model. Modifying the underlying differential equations - e.g., adding a physiological state or altering an epidemiological contact network - renders trained models obsolete and forces computationally expensive retraining from scratch. We introduce HyperODE, a surrogate capable of operating across an entire class of approximately mass-conserving compartmental models without retraining. By mapping the structure of ordinary differential equations (ODEs) into directed hypergraphs, HyperODE decouples the functional form of system interactions from the neural network architecture. HyperODE takes a compartmental model in the form of an ODE with an arbitrary parameter distribution defined through quantiles and transforms it into a hypergraph. It outputs the distribution of the trajectories for all the states in the original ODE in the form of quantiles. We then use this surrogate to build an encoder that takes a noisy trajectory and outputs a distribution over the parameters of the original ODE, thus calibrating the model in a single pass. On families and system sizes never seen in training, HyperODE produces calibrated quantile bands in a single forward pass, with weighted-interval score and coverage on par with specialized surrogates for each structure. For inverse inference, HyperODE produces calibration from noisy state trajectories in a few milliseconds with a single shared encoder, competitive with existing methods. HyperODE extends zero-shot to ODEs that break mass conservation and to external forcing.
Ajitesh Srivastava· 0 citations
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