# CryptoL: Towards Scale Dominance and Physics Constraints Mitigation in Financial Multivariate Time Series Forecasting

arXiv: https://arxiv.org/abs/2609.11206 · HTML: https://mlsanity.com/paper/2609.11206/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 5 of its 18 references that are in the map.
- First posted on arXiv 10 Sep 2026.

## Builds on (most cited first, 5 of 5)

- [Time-MoE: Billion-Scale Time Series Foundation Models with Mixture of Experts](https://mlsanity.com/paper/2409.16040/) (2024)
- [Only the Curve Shape Matters: Training Foundation Models for Zero-Shot Multivariate Time Series Forecasting through Next Curve Shape Prediction](https://mlsanity.com/paper/2402.07570/) (2024)
- [On the Role of Reversible Instance Normalization](https://mlsanity.com/paper/2603.11869/) (2026)
- [Multivariate Financial Forecasting using the Chronos Time Series Foundation Models](https://mlsanity.com/paper/2605.21504/) (2026)
- [Autoencoder based Hybrid Multi-Task Predictor Network for Daily Open-High-Low-Close Prices Prediction of Indian Stocks](https://mlsanity.com/paper/2204.13422/) (2022)

## Abstract (arXiv)

Cryptocurrency forecasting presents a distinctive combination of extreme cross-asset scale heterogeneity, non-stationary dynamics, and structural dependencies among Open, High, Low, and Close (OHLC) variables. We present CryptoL, a unified framework designed to address these challenges within multivariate time-series forecasting. CryptoL evaluates forecasting error in context-normalized coordinates within the RevIN pipeline, preventing inverse normalization from introducing an additional squared-scale weighting into the MSE objective. We formally characterize this effect through the empirical risk and parameter-gradient geometry, establishing the conditions under which large-scale assets can disproportionately influence shared-model optimization. Beyond loss-space normalization, CryptoL examines channel-independent and channel-dependent normalization for OHLC data, showing that a shared channel-dependent affine transformation preserves candle-order relations that independent channel transformations need not preserve. The framework further incorporates scale-adaptive numerical stabilization to reduce distortions caused by a fixed normalization constant across assets spanning many orders of magnitude, together with a soft feasibility loss that penalizes violations of the defining OHLC inequalities. Experiments across heterogeneous cryptocurrency assets evaluate these components through controlled ablations and demonstrate improvements in forecasting accuracy, training stability, and the frequency of financially valid OHLC predictions relative to the considered baselines. CryptoL therefore provides an integrated approach to scale-balanced optimization, structure-preserving normalization, numerical stabilization, and constraint-aware cryptocurrency forecasting.