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

Yalda Taheri; Mohammad Hassan Heydari; Armon Rasooli; Maryam Amirshahkarami; Mohammad Ebrahim Mahdavi; Hossein Karshenas

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

Yalda Taheri, Mohammad Hassan Heydari, Armon Rasooli, Maryam Amirshahkarami, Mohammad Ebrahim Mahdavi, Hossein Karshenas

arXiv:2609.11206v1Sep 10, 2026cs.AIcs.CEcs.LG

TL;DR

Cryptocurrency forecasting must handle scale heterogeneity, non-stationarity, and OHLC structural dependencies simultaneously. CryptoL combines normalized-space optimization, structure-preserving normalization, adaptive stabilization, and a soft constraint loss, with experiments reporting improved accuracy, stability, and physical validity across heterogeneous assets.

Problem
Cryptocurrency multivariate forecasting must jointly address extreme cross-asset scale heterogeneity, non-stationary dynamics, and structural dependencies among OHLC variables.
Method
CryptoL combines channel-aware normalization, TP-RevIN normalized-space optimization, scale-adaptive epsilon stabilization, and a soft OHLC feasibility loss.
Results
CryptoL improves forecasting accuracy, training stability, and the frequency of financially valid OHLC predictions relative to the considered baselines.
Takeaways & Limitations
The integrated framework offers a practical compromise between forecasting accuracy and physical consistency for shared models trained on heterogeneous financial time series.
Takeaways & Limitations
TP-RevIN removes scale-induced weighting but does not guarantee equal gradient norms or equal difficulty across assets because forecastability, residual structure, sampling frequency, and model sensitivity still differ.

Abstract

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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.

Introduction

CryptoL addresses the combined challenges of non-stationarity, extreme cross-asset scale heterogeneity, and structurally invalid OHLC forecasts in cryptocurrency markets. It integrates structure-aware normalization, scale-balanced optimization, numerical stabilization, and a physics-informed loss.

Introduction: Cryptocurrency OHLC forecasting is difficult because abrupt regime changes, non-stationarity, cross-asset scale heterogeneity, and strict candlestick rules must be handled jointly.Assets can range from fractions of a cent to tens of thousands of dollars, complicating shared-model optimization.
Introduction: CryptoL combines structure-aware normalization, TP-RevIN, dynamic epsilon stabilization, and an auxiliary physics-informed loss for heterogeneous cryptocurrency forecasting.The framework is designed to balance optimization across asset scales while discouraging invalid candlestick structures and maintaining numerical stability.
Introduction: Even normalized forecasting models can produce financially inadmissible predictions that violate core OHLC inequalities.CryptoL addresses this structural problem with an auxiliary loss that penalizes invalid candlestick configurations.
Introduction: The study compares channel-independent and channel-dependent RevIN, fixed and scale-adaptive epsilon formulations, TP-RevIN, and alternative normalization approaches.These analyses examine candlestick structure, numerical stability, scale-induced optimization imbalance, and constraint enforcement.

Related Work

Prior work has studied multivariate financial forecasting, OHLC structural validity, and normalization for non-stationary time series. CryptoL builds on these directions by jointly considering OHLC-aware normalization, normalized-space optimization, adaptive stabilization, and constraint enforcement.

Related Work: Financial forecasting research has progressed from classical state-space and correlation models to recurrent, autoencoder, Transformer, MLP, and foundation-model approaches.These methods aim to model temporal, cross-variable, and cross-asset dependencies jointly.
Related Work: Existing OHLC forecasting methods enforce valid outputs through invertible transformations or model channel dependencies with hybrid autoencoder and multitask architectures.Other approaches incorporate OHLC event timestamps to enrich bar representations.
Related Work: RevIN, SAN, and FAN address distribution shifts through instance, local temporal-slice, or frequency-based normalization strategies.These methods primarily target temporal non-stationarity rather than the normalization axis for structured multivariate variables.
Related Work: CryptoL jointly studies OHLC-aware normalization, normalized-space optimization, adaptive numerical stabilization, and auxiliary enforcement of candlestick constraints.This combination extends prior work beyond treating temporal distribution changes alone.
Related Work: Recent studies distinguish representation normalization from the coordinate system used to evaluate training loss.GTT trains on context-normalized targets, while comparative analyses report that denormalized MSE and MAE remain scale sensitive.

Methodology

CryptoL combines structure-aware OHLC normalization, scale-balanced TP-RevIN training, adaptive stabilization, and normalized-space constraint losses for heterogeneous cryptocurrency forecasting.

OHLC normalization: Channel-dependent normalization preserves relative OHLC organization through one shared affine transformation, whereas channel-independent normalization does not mathematically guarantee the original ordering.CI standardizes channels separately; CD computes shared statistics and applies the same transformation across OHLC channels.
Numerical stabilization: Dynamic epsilon adapts stabilization to context magnitude, addressing disproportionate effects from fixed constants on very small cryptocurrency prices.The formulation is designed to prevent a fixed numerical constant from distorting normalization for low-valued assets.
TP-RevIN and scale balance: TP-RevIN evaluates forecasting and OHLC constraint losses in normalized coordinates, applying inverse normalization only during inference and reporting.This prevents the auxiliary constraint loss from reintroducing the scale weighting removed by TP-RevIN.
TP-RevIN and scale balance: TP-RevIN removes the explicit scale-dependent weighting that original-space RevIN introduces into the empirical objective and parameter updates.For comparable normalized errors and sensitivities, original-space updates scale with squared context scales, whereas TP-RevIN removes this multiplicative weighting.
Scope and assumptions: The framework targets cryptocurrency panels where assets span many orders of magnitude, while retaining differences in forecastability, residual structure, sampling, and model sensitivity.TP-RevIN removes only weighting introduced by RevIN scale and does not guarantee equal difficulty or equal gradient norms across assets.
Unified framework: CryptoL combines CI or CD normalization, dynamic epsilon, TP-RevIN, and normalized-space OHLC constraints into a unified forecasting pipeline.The backbone predicts in normalized coordinates, both losses are optimized there, and predictions are restored to physical scale afterward.

Experiments

CryptoL is evaluated on a large, heterogeneous cryptocurrency dataset using multiple decoder-only forecasting backbones and joint training across 16 assets. The experimental setup uses distributed TPU training and reports metrics after restoring predictions to physical space.

Dataset: The dataset contains approximately 15.5 million OHLC observations from 16 cryptocurrency assets across multiple temporal resolutions and forecast horizons.Asset prices range from approximately 10^-7 to above 10^5, creating substantial cross-asset scale heterogeneity.
Models and training: CryptoL is evaluated with Timer, Timer-XL, and Time-MoE decoder-only forecasting backbones using AdamW and a learning rate initialized at 10^-5.Each model is trained for one epoch using 90% of the data, with 10% reserved for evaluation.
Evaluation protocol: Forecasting metrics are computed after restoring predictions to the original physical space for consistent comparison across normalization and training configurations.The evaluation split contains the remaining 10% of available data.
Evaluation protocol: Each configuration trains jointly on all 16 assets to test whether shared backbones learn transferable temporal and cross-asset patterns despite differences in scale, volatility, and market behavior.Experiments are conducted separately by time frame and forecast horizon.
Implementation: The experiments use eight TPU v5e accelerators with fully sharded data-parallel training to distribute parameters, optimizer states, and input data.The implementation uses JAX and Flax, with model parameters and intermediate activations represented in FP32.

Results and Analysis

The results show that TP-RevIN improves scale-balanced forecasting across heterogeneous cryptocurrency assets, dynamic epsilon stabilizes low-value assets, and auxiliary physics loss balances accuracy with physical validity.

Normalization strategies: Without RevIN, models produce extremely high MSE values because of immense scale discrepancies among cryptocurrency assets.Standard CI and CD RevIN mitigate this problem, while TP-RevIN yields further reductions.
Normalization strategies: TP-RevIN consistently reduces MSE and MAE across forecasting horizons and backbones, while FAN and SAN produce substantially higher MAE and MAPE values.TP-RevIN also maintains lower physical violation rates than the compared temporal normalization schemes.
Numerical stabilization: Dynamic epsilon reduces SHIB MAPE to 1.87 / 3.51 under TP-RevIN, whereas fixed epsilon produces a MAPE of 17362.6 for Time-MoE under standard RevIN.Fixed epsilon has negligible impact for BTC and ETH but severely distorts normalization for low-value SHIB.
Numerical stabilization: The fixed-versus-dynamic epsilon analysis evaluates numerical stabilization across assets with different price scales using Table 3.The comparison specifically includes high-value BTC and ETH and low-value SHIB.
Physics-informed constraint loss: Auxiliary physics loss reduces candlestick violations to near-zero levels while keeping TP-RevIN MAE highly competitive.The unconstrained-space method guarantees PHY = 0.0 but can yield MAE of 3307.53 versus 34.77 for TP-RevIN (CI) on Timer-XL at 30m and horizon 5.

Conclusion

CryptoL integrates normalized-space optimization, OHLC-aware normalization, adaptive stabilization, and a soft structural loss for heterogeneous cryptocurrency forecasting. Its design targets both scale-balanced learning and physically valid candlestick predictions.

Conclusion: CryptoL combines Two-Phase RevIN, dynamic epsilon stabilization, and normalized-space auxiliary physics loss to address scale heterogeneity, numerical instability, and candlestick violations.The framework is presented as a practical compromise between forecasting accuracy and physical consistency.
Structure-aware normalization: Channel-dependent normalization preserves OHLC ordering under a shared positive affine transformation, whereas independent channel transformations do not guarantee that preservation.Under channel-dependent normalization, normalized-space OHLC constraints are exactly equivalent to original-space constraints.
Scale-balanced optimization: TP-RevIN evaluates forecasting error directly in normalized target coordinates, avoiding squared context-scale weighting introduced by inverse normalization before MSE computation.This weighting can alter aggregate gradient direction and local optimization curvature in shared multi-asset models.
Constraint-aware forecasting: The auxiliary OHLC consistency loss is evaluated in normalized coordinates and substantially reduces original-space violations under both channel-independent and channel-dependent normalization.Under channel-independent normalization, the loss acts as a representation-space regularizer and financial validity remains evaluated after inverse normalization.
Numerical stabilization: Dynamic epsilon scales with the squared context mean, reducing the disproportionate influence of a fixed stabilizer on extremely low-valued assets.The formulation is approximately equivariant under positive changes of numerical units and makes regularization depend primarily on relative scale.

B Mathematical Analysis

The analysis proves that shared positive affine normalization preserves OHLC ordering, whereas channel-independent normalization does not guarantee it. It also characterizes the resulting difference for normalized-space physics constraints.

Order preservation under CD normalization: Shared positive affine normalization preserves every valid OHLC ordering after transformation.
Physics-loss equivalence: Under channel-dependent normalization, the normalized physics-loss zero set is identical to the corresponding raw-space constraint zero set.
Channel-independent normalization: Channel-independent normalization can reverse a valid OHLC relation, so original validity does not imply normalized-space validity.
Physics-loss interpretation: Under channel-independent normalization, the physics loss acts as a representation-space regularizer and raw-space validity must be checked after inverse normalization.

Mathematical Analysis of Dynamic Epsilon

The dynamic-epsilon analysis contrasts fixed stabilization, which is not scale-equivariant, with a context-scale-based formulation that is approximately scale-equivariant. Its regularization effect depends mainly on relative variability but can dominate nearly constant contexts.

Fixed epsilon: Fixed-epsilon normalization is non-equivariant because its denominator depends on absolute numerical scale.
Dynamic epsilon: Dynamic epsilon is approximately scale-equivariant under positive rescaling when its additive term is sufficiently small.
Relative variability: For assets with comparable relative volatility, dynamic epsilon has an effect that is approximately independent of absolute quoted price.
Limitation: Dynamic epsilon may dominate empirical variance and compress variations in nearly constant contexts, requiring empirical checks across volatility regimes.

Jacobian Analysis of Scale-Dominance Mitigation

The Jacobian analysis shows that inverse-normalized MSE introduces explicit scale weighting into gradients and curvature, while TP-RevIN removes that weighting. The result is scale neutrality rather than a universal guarantee of better conditioning.

Gradient scaling: Original-space RevIN weights equal normalized residual-Jacobian contributions in proportion to the squared context scale, whereas TP-RevIN removes this explicit multiplier.
Scope of gradient claim: The scale-weighting result does not imply that every high-scale sample has a larger gradient because residuals, Jacobians, sampling, and task difficulty also matter.
Jacobian conditioning: Under orthogonal asset subspaces, TP-RevIN removes scale-induced curvature weighting, while original-space RevIN produces singular-value ratios tied to smax/smin.
Scope of conditioning claim: TP-RevIN does not universally minimize nonlinear Jacobian condition numbers because intrinsic anisotropy and compensating scale effects may remain.

C Dataset Details and Volatility Analysis

The dataset contains 15,490,300 OHLC rows from 16 heterogeneous assets across seven temporal resolutions. Volatility rises sharply with timeframe, making longer-horizon forecasting a more difficult regime.

Dataset composition: 15,490,300 OHLC rows span 16 assets, seven temporal resolutions, and 112 distinct time-series panels.
Asset coverage: The assets include mature coins with histories dating to 2017 and newer tokens representing lower-history, high-volatility regimes.
Volatility analysis: Daily candles show 37.46% exceeding a 1% change and 11.33% exceeding 5%, versus 0.99% and 0.01% at five-minute resolution.
Optimization difficulty: Forecasting accuracy declines on larger timeframes as extreme returns increase target-distribution variance.

D Experimental Details

The experiments use controlled normalization, stabilization, and physical-consistency comparisons across specified timeframes, horizons, and training settings. Dataset tables report observed rows and volatility distributions partitioned by asset and timeframe.

Normalization comparisons: Experiments compare models with no normalization, standard RevIN, and TP-RevIN under MSE-only training across 5m and 30m timeframes and forecast horizons of 5 and 15 steps.The input context window is L = 480, λ_phy = 0, and the normalization constant is fixed at ϵ = 10^-5.
Dataset organization: Table 6 records observed data row counts partitioned by coin and timeframe.The table provides the dataset-count basis for the experimental partitions.
Volatility analysis: Table 7 reports aggregate volatility distributions across all assets, while Tables 8 and 9 provide granular thresholds for the evaluated cryptocurrency assets.These tables organize volatility evidence across experimental resolutions and asset groups.
Baseline comparisons: The alternative-normalization comparison evaluates FAN, SAN, and TP-RevIN on the 1h timeframe with a 480-step lookback and MSE-only training.FAN uses joint optimization, SAN uses a two-stage schedule, and TP-RevIN uses a one-epoch scale-adaptive dynamic-epsilon configuration.
Numerical stabilization: The stabilization study compares fixed and dynamic epsilon on the 2h timeframe to isolate denominator-regularization effects across low- and high-valued assets.The evaluation uses L = 480 and the MSE objective, contrasting assets such as SHIB and BTC.
Physical consistency: Physical-consistency evaluations use L = 480 across 30m and 1h intervals, with auxiliary candlestick losses evaluated in original physical coordinates under standard RevIN.All evaluations are conducted in original space for fair comparison.