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Signals, Concepts, and Laws: Toward Universal, Explainable Time-Series Forecasting

Hongwei Ma, Junbin Gao, Minh-Ngoc Tran

TL;DR

DORIC addresses the challenge of accurate, explainable forecasting for heterogeneous multivariate time-series by introducing a universal five-concept bottleneck and a physics-informed head based on a driven–damped ODE. The model enforces data fidelity, concept alignment, and physical plausibility through a joint loss, enabling interpretability without sacrificing cross-domain performance. Theoretical results guarantee universal expressiveness and convergence under a physics ramp-up, while extensive experiments across six benchmarks show competitive accuracy and robust interpretability. This approach yields a transferable, physically grounded forecasting framework suitable for safety-critical and cross-domain applications.

Abstract

Accurate, explainable and physically credible forecasting remains a persistent challenge for multivariate time-series whose statistical properties vary across domains. We propose DORIC, a Domain-Universal, ODE-Regularized, Interpretable-Concept Transformer for Time-Series Forecasting that generates predictions through five self-supervised, domain-agnostic concepts while enforcing differentiable residuals grounded in first-principles constraints.

Signals, Concepts, and Laws: Toward Universal, Explainable Time-Series Forecasting

TL;DR

DORIC addresses the challenge of accurate, explainable forecasting for heterogeneous multivariate time-series by introducing a universal five-concept bottleneck and a physics-informed head based on a driven–damped ODE. The model enforces data fidelity, concept alignment, and physical plausibility through a joint loss, enabling interpretability without sacrificing cross-domain performance. Theoretical results guarantee universal expressiveness and convergence under a physics ramp-up, while extensive experiments across six benchmarks show competitive accuracy and robust interpretability. This approach yields a transferable, physically grounded forecasting framework suitable for safety-critical and cross-domain applications.

Abstract

Accurate, explainable and physically credible forecasting remains a persistent challenge for multivariate time-series whose statistical properties vary across domains. We propose DORIC, a Domain-Universal, ODE-Regularized, Interpretable-Concept Transformer for Time-Series Forecasting that generates predictions through five self-supervised, domain-agnostic concepts while enforcing differentiable residuals grounded in first-principles constraints.

Paper Structure

This paper contains 57 sections, 2 theorems, 36 equations, 6 figures, 12 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Contributions.
  3. Scope of interpretability and scientific value.
  4. Literature Review
  5. Methodology
  6. Raw Data and Forecasting Goal
  7. The Causal Transformer Encoder
  8. Initial Token Embedding:
  9. Masked Multi-Head Attention:
  10. Latent History Vector:
  11. The Concept Bottleneck Layer $g_{\phi}$
  12. Learning Concepts:
  13. Analytical Soft Targets:
  14. Notation for concepts and soft targets.
  15. Concept Alignment Loss:
  16. ...and 42 more sections

Key Result

Theorem 1

Assume $f^\star$ is continuous on $K$ and its latent dynamics satisfy equation (eq:ode). Then for every $\varepsilon>0$ there exist parameters $\Theta$ and an embedding width $d$ such that

Figures (6)

  • Figure 1: DORIC Structure
  • Figure 2: Concept Correlation Heatmap.
  • Figure 3: DORIC's Predictions. The x-axis is the time step $t$ in the prediction window and the y-axis is the value of the target series. Blue: ground truth; Orange: DORIC
  • Figure 4: Concept trajectories and analytic targets on six datasets. For each benchmark (Electricity, Traffic, Weather, Illness, Exchange rate, ETT), we plot a representative channel together with the learned concepts $c_{k,t}$ and their analytic targets $c_{k,t}^{\ast}$. Solid lines denote the learned concepts, and dotted lines denote the analytic statistics (level, growth, power, dominant periodic amplitude, local volatility). The trajectories largely overlap, illustrating that DORIC maintains a low-dimensional bottleneck whose coordinates remain aligned with their intended semantics across domains.
  • Figure 5: Gradient alignment between physics and data losses. We track the gradient-alignment statistic $\kappa_t = \frac{1}{5}\sum_{k=1}^5 \cos( \nabla_{c_{k,t}} L_{\mathrm{phys}}, \nabla_{c_{k,t}} L_{\mathrm{data}} )$ over epochs for several datasets. After an initial transient, $\kappa_t$ becomes non-negative and stabilises, indicating that the physics residuals and data loss exert compatible pressures on the concepts rather than fighting each other. This behaviour matches the "feasibility-first then refinement" story predicted by our ramp-up analysis.
  • ...and 1 more figures

Theorems & Definitions (4)

  • Theorem 1: Universal Expressiveness of DORIC
  • proof
  • Theorem 2: SGD with Physics Ramp-up
  • proof