Exploratory Modelling of Multi-System Transformation Pathways from Real-World Data: A SINDy-Inspired Sparse Orthogonal Regression Technique

Technical Summary: Exploratory Modelling of Multi-System Transformation Pathways

Problem Statement
Sustainability transitions involve interacting dynamics across social, technological, economic, environmental, and governance dimensions. Existing modelling approaches often face a dichotomy: they either isolate subsystems or rely on optimization-based pathways that fail to explicitly represent feedbacks, path dependence, and institutional constraints. Furthermore, traditional models struggle to capture the non-linear, asynchronous, and often crisis-driven nature of socio-ecological transitions while maintaining transparency and empirical grounding. There is a need for a modelling framework that bridges quantitative dynamical systems and transition theory without imposing rigid structural assumptions or claiming deterministic forecasting precision.

Methodology: Sparse Orthogonal Regression Technique (SORT)
The study introduces the Sparse Orthogonal Regression Technique (SORT), a data-driven dynamical-systems framework inspired by the Sparse Identification of Non-linear Dynamical Systems (SINDy). Unlike approaches that specify a full structural model ex ante, SORT infers parsimonious cross-domain dependencies directly from harmonized time-series data.

• Data and Variables: The prototype utilizes a 20-year empirical window (2005–2025) covering EU-wide indicators across ten domains: renewable energy share, ETS-regulated emissions, resource productivity, digitalisation intensity, clean innovation (green patents), policy stringency, transition finance, social well-being, environmental stress, and system resilience.
• Algorithmic Approach: The method constructs a library of potential explanatory terms (linear and low-order non-linear interactions) from the observed variables. These terms are represented through an orthogonal basis expansion to ensure stable and compact approximations. Sparse regression is then applied to select the subset of terms that best explains the observed rates of change, penalizing model complexity to avoid overfitting.
• Philosophy: The resulting compact dynamical system is interpreted as a structural representation of medium-term interactions and feedback loops, not as a forecasting tool. The approach prioritizes identifying dominant dependencies and sign-consistent influences over precise numerical prediction.

Key Results
The study compares the SORT-derived dynamical system (DS) against simple univariate linear extrapolations across the ten variables:

• Divergence from Linear Trends: While the DS model closely reproduces historical data, its forward projections (2025–2045) diverge from linear trends where reinforcing feedbacks exist.

  • ETS Emissions: The DS model projects a more pronounced decline than linear extrapolation, suggesting that under aligned structural conditions (finance, productivity, stress), decarbonisation can accelerate non-linearly.
  • Transition Finance: The model captures a convex acceleration in green finance, reflecting endogenous reinforcement as coordination mechanisms mature, whereas linear models assume constant marginal growth.
  • Resource Productivity: The DS model shows an acceleration in productivity gains driven by coordinated technological upgrading, contrasting with the steady rate assumed by linear trends.
  • Environmental Stress: Unlike linear models that assume steady accumulation, the DS model captures non-linear oscillations and conditional stabilization, reflecting the hydrological and systemic nature of stress.
  • Stability: Variables such as renewable energy share, digitalisation, and policy stringency show trajectories where the DS model remains close to linear extrapolation, indicating structural stability or slow-moving institutional rhythms.

• Inferred Causal Structure: The sparse regression reveals a compact set of recurring dependencies, organized into three interacting domains:

  1. Social-Technological: A mutually reinforcing feedback loop between clean innovation and social well-being.
  2. Environmental-Productive: Environmental stress acts as a constraining hub, negatively influencing both renewable energy uptake and resource productivity.
  3. Governance-Transition: Policy stringency supports financial mobilisation, which in turn enables innovation dynamics. Resilience (financial integration) acts as a stabilizing modifier rather than a direct driver.

Significance and Claims
The paper positions SORT as a "causal scaffolding" tool for science-policy interfaces rather than a predictive engine. Its primary contributions are:

1. Empirical Grounding of Transition Theory: The model provides an empirically grounded representation of multi-system feedback dynamics, bridging the gap between abstract transition theory and quantitative modelling. It demonstrates that sustainability transitions are co-evolutionary processes where technological change, governance, finance, and social conditions are structurally interdependent.
2. Exploratory Reconstruction: The technique offers a compact method for reconstructing transformation pathways from real-world data, identifying structural tendencies and conditional dynamics (e.g., how accelerated decarbonisation is contingent on the alignment of finance, innovation, and stress reduction).
3. Policy Relevance: By translating inferred dependencies into feedback structures, the analysis identifies reinforcing sequences (e.g., innovation-wellbeing) and balancing constraints (e.g., environmental stress). This supports the formulation of policy as structural steering principles rather than optimization targets, emphasizing the need for cross-domain coordination to activate positive feedbacks and mitigate destabilizing couplings.

The authors maintain a modest stance, acknowledging that the model is a feasibility study and a prototype. They assert that the value lies in the interpretability of the causal structure and the generation of transparent counterfactuals, rather than in forecasting deterministic futures. The work serves as a foundation for developing more elaborate versions and integrating these data-driven insights with participatory scenario planning.