Universal Differential Equations for Stable Multi-Step Volatility Time Series Forecasting
Authors: Prasanna Devadiga
Kishan Gurumurthy
Kshitij Mohan
Venue: Accepted by TMLR ()

Abstract: Neural differential equations such as Neural ODEs, Neural CDEs, and Universal Differential Equations (UDEs) model temporal evolution as a continuous-time flow rather than a fixed-step recurrence. Even for regularly sampled data, this formulation differs fundamentally from discrete-time architectures: it learns smooth vector fields governing instantaneous rates of change, reducing discretization bias and improving long-horizon stability. We present a systematic study of Universal Differential Equations for financial volatility forecasting, a domain characterized by regime shifts, heavy tails, and jump discontinuities. UDEs extend Neural ODEs by embedding mechanistic structure within learned dynamics, using neural networks to parameterize coefficients in partially known differential equations instead of learning the system purely from data. Our UDE variants incorporate volatility’s empirical regularities while retaining neural flexibility for regime adaptation. Across market regimes, they outperform both continuous-time baselines and discrete-time models, achieving higher accuracy and greater long-horizon stability while remaining interpretable. These results suggest that UDEs grounded in mechanistic structure and neural flexibility offer a principled route to stable, interpretable multi-step forecasting in nonstationary domains.

Published in Transactions on Machine Learning Research (12/2025)
Universal Differential Equations for Stable Multi-Step Volatility Time Series Forecasting
Prasanna Devadiga
dewdiioopeeennno288@maaa.co com
TIPIN
Kishan Gurumurthy TIFIN
keshanurrmmarthyy@@maaa.com
Kshitij Mohan TIPIN
kshtttmmohhn//5@mmaal.cmmm
Reviewed on OpenReview:
https: // openreview. net/ forum? id= uWGNemco2f
Abstract
Neural differential equations such as Neural Ordinary Differential Equations (Neural ODFs), Neural Controlled Differential Equations (Neural CDEs), and Universal Differential Equa- lions (UDEs) model temporal evolution as continuous-time flow rather than a fixed-step recurrence. Even for regularly sampled data, this formulation differs fundamentally from discrete-time architectures: learns smooth vector fields governing instantaneous rates of change, reducing discretization bias and improving long-horizen stability. We present sys- tematic study of Universal Differential Equations for financial volatility forecasting, domain characterized by regime shifts, heavy tails, and jump discontinuities. UDEs extend Neural ODEs by embedding mechanistic structure within learned dynamics, using neural networks to parameterize coefficients in partially known differential equations instead of learning the system purely from data. Our UDE variants incorporate volatility's empir- ical regularities while retaining neural flexibility for regime adaptation. Our formulation approximates the aggregate impact of jumps through smooth, continuous dynamics rather than explicit stochastic arrivals, relaining tractability and interpretability. Across market regimes, they outperform both continuous-ting time baselines and discrete time models, achiev- ing higher accuracy and greater long-horizon stability while remaining interpretable. These results suggest that UDEs grounded in mechanistic structure and neural flexibility offer a principled route to stable, interpretable multi-step forecasting in nonstationary domains.
1
Introduction
Time series forecasting is fundamentally challenging due to complex temporal dependencies that make ac- curate prediction difficult (25 19.. These challenges are particularly acute in domains where the data- generating process itself is unstable and where small modeling errors can have cascading consequences.
We focus on financial volatility as our evaluation domain: Volatility serves as demanding testbed for three reasons. First, it is practically important: volatility quantifies market risk and is fundamental to portfolio construction and systemic risk management (3) IT]. Second, open data (from yfimance) enables reproducible research without computational burden. Third, volatility exhibits extreme versions of forecast- ing challenges through well-documented 'stylized facts' (6): clustering (high-volatility periods persist), jump discontinuilies (sudden spikes from large price movements), and heavy-lailed distributions (extreme events occur more frequently than normal distributions predict). These properties make even single-step prediction difficult, as models trained during calm markets fail catastrophically under regime shifts. While volatility dynamics are often modeled with stochastic differential equations (SDEs) incorporating explicit Pois- son jump terms, such formulations introduce heavy computational and identifiability burdens. In this work.

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Published in Transactions on Machine Learning Research (12/2025)
we instead approximate jump effects through a smooth deterministic modulation within contimuuus-tmme ODE framework, preserving differentiability and solver stability while retaining the key phenomenological behavior of sudden volatility bursts.
Multi-step forecasting amplifies these challenges: Errors compound with horizon, subtle mis- specifications become catastrophic failures, and models must maintain stability while recursively feeding their own predictions forward. For volatility, clustering means one error during high-volatility regimes can destroy entire forecast trajectories-1f the model underestinates at step t, it continues underestimating ++ 1, + 2.... as clustering persists. Unlike smoother time series where multi-step errors degrade gradually, volatility forecasts collapse abruptly at longer horizons.
Current approaches fall short in different ways: This motivates hybrid approaches combining neu- ral flexibility with mechanistic stability. Fully data-driven architectures (LSTMs, GRUs, Transformers) learn discrcto-time transitions as black-box functions, roquiring massive datascts and often failing at long horizons (25 Iv] Neural ODEs improve inductive bias through continuous-tine dynamics but still learn unconstrained vector fields from data (5). Mechanistic models (GARCH, Heston SDEs) encode domain knowledge but cannot adapt beyond fixed parametric forms (II ( and are computationally expensive to train when extended to full stochastic (SDE or jump-diffusion) formulations. UDEs bridge this gap by embedding mechanistic structure while allowing neural modulation of key coefficients.
Universal Differential Equations bridge this gap: An Ordinary Differential Equation (ODE) specifies a fully known righthand side, ズ = fa,,t; i);; a Universal Differential Equation (UDE) augments a mechanistic ODE with learned component, & /xoown (zr, 1; 0) (NN(2,,;; or equivalently learns state-dependent coefficients while retaining the governing form. In this paper, we adopt the modulation view: small MLPs modulate the mean reversion coelficients under positivity/boun /bound constraints to preserve dissipativily and interpretability, in contrast to a vanilla neural ODE that learns an unconstrained vector field.
UDEs for forecasting: A different problem class: While neural ODEs and SDEs have been applicd to various forecasting lasks, UDEs were introduced for system identification and scientific discovery, where models fit to complete observed trajectories with ground-truti feedback at each step (2I). Multi-step fore- casting presents fundamentally different challenges: models must maintain stability under autoregressive rollout, recursively feeding their own predictions forward without ground-truth correction. This autoregres- sive structure causes errors to compound over time. A small bias at horizon h 1 becomes catastrophic degradation at / = 20 as the model repeatedly consumes its own outputs. To our knowledge, we provide the first systematic study demonstrating that UDEs, when properly constrained through compositional stacking of domain structure, remain stable under this auloregressive stress test, where unconstrained neural ODFs collapse (Section 2
Why not SDEs Neural SDEs? Stochastic differential equations provide principled way to represent uncertainty, but neural jump-augmented SDEs are computationally expensive and often numerically unstable. Training requires stochastic integration and Monte-Carlo gradient estimation at every step, Icading to high variance and long runtimes. Moreover, diffusion and jump parameters are difficult to identify jointly from historical volatility alone without exogenous information (options data, macro factors). To maintain tractability and interpretability, we focus on deterministic UDEs that embed financial structure while remaining computationally officient and stable under long rollouts.
Our approach: Compositional UDEs for volatility: We use neural networks to modulate mechanis- tic coefficients within governing differential equation rather than learning dynamics from scratch. For volatility forecasting specifically, we incorporate mean-reversion dynamics with task-specific coefficients: mean-reversion rates (speed of return to equilibrium) and long-term volatility levels. This preserves the stabilizing inductive bias of the mechanistic form while allowing adaptation to regime-dependent dynamics. The result is a model that maintains interpretability each learned coefficient corresponds to a meaning- ful market quantity while achieving strong multi-step stability. Our framework leverages volatility's stylized facts as domain knowledge while allowing neural components to learn continuous, state-dependent parameter variations. The mechanistic structure constrains dynamics to remain mean-reverting by construction, while neural modulation enables smooth adaptation to evolving market conditions within that stable template.
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Model hierarchy and terminology: Our experimental framework spans spectrum from pure data- driven to pure mechanistic models. At one extreme, Vanilla Neural ODE learns dynamics entirely from data with no structural priors. At the other, ODE-MR is a pure mean-reverting process with fixed pa- rameters. Between these extremes, our UDE variants progressively incorporate stylized facts: UDE-MR adds neural modulation of mean-reversion coefficients; UDE-MRVC further adds volatility clustering; and UDE-MRVCJ additionally incorporates jump dynamics through a smooth deterministic modulation term that approximates average jump effects within a continuous-tine formulation. All UDE variants enforce soft constraints, such nonnegative mean-reversion rates, bounded long-run levels, and structured clustering terms to ensure neural components respect rather than overturn the mechanistic structure.
Our contributions are:
1. Application of UDEs to autoregressive multi-step forecasting: Unlike previous UDE appli- cations in system identification where models fit complete observed trajectories, multi-step forecast- ing requires stability under repeated self-feeding of prodictions without ground truth correction. We demonstrate that structurally constrained UDEs maintain performance under this autoregressive stress test, using rigorous rolling walk-forwarr validation across asset classes.
2. Parameter efficiency through structural inductive biases: UDEs achieve competitive accu- racy with significantly fewer parameters than foundation models, demonstrating that embedding domain structure can replace massive scale for forecasting tasks with partial mechanistic knowledge.
Interpretable regime adaptation: Unlike black-box foundation models, the hybrid architecture maintains interpretability through its mechanistic structure. Learned parameters correspond to economically meaningful quantities (mean-reversion rates, volatility clustering strength, jump inten- sities) that can be tracked across time, enabling post-hoc analysis of how market dynamics evolve across regimes (Figure 2).
Long-horizon stability: UDEs remain relatively stable across longer horizons, unlike purely neural and foundation models, which degrade at higher rale.
All results are reported in trading days, where 1, 5, 10, 20 steps correspond approximately to daily, weekly, bi-weekly and monthly time-periods periods for most assets.
2 Related Work
Multi-step Time Scries Forecasting: Multi-step forecasting presents unique challenges due to error accumulation and distribution shift, with approaches broadly categorized into direct, recursive, and multi- output strategies
(23). Deep learning architectures have significantly advanced the field, including L.SNNtt (I7) which combined CNNs and RNNs for temporal patterns, and transformer-based methods like Informer 28 and Autoformer (5 that capture long-term dependencies through specialized attention mechanisms.
Recent slate-of-the-art foundation models like TimesFM { leverage large-scale pretraining across diverse temporal domains to enable robust, zero-shot multi-step forecasting.
Neural Differential Equations for Time Series: Neural ordinary differential equations (Neural ODEs) 5 model temporal evolution as continuous dynamical systems, enabling natural interpolation and extrapolation. They have been applied to various forccasting domains including energy demand prodic- tion (28), weather forecasting (24), and ecological population dynamics (2). Neural stochastic differential equations (Neural SDEs) extend this framework by incorporating stochastic terms for uncertainty quantifica- tion, finding applications in retail sales forecasting and financial modeling (9) Neural controlled differential equations (Neural CDEs) (66 have been evaluated on standard time series benchmarks, demonstrating advantages for sremulerly-samplel data. However, these works primarily focus on system modeling or short- horizon prediction tasks, with evaluation protocols typically using limited validation strategies.
Volatility Forecasting: Financial volatility forecasting has evolved from parametric econometric models flexible machine learning approaches. The GARCH family 3 12 and stochastic volatility models (13 have
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dominated academic finance, providing interpretable but potentially misspecified parametric forms. Recent machine learning approaches include neural networks for volatility prediction (27), deep learning models capturing complex patterns (4, and hybrid methods that combine GARCH with neural networks as feature extractors (18).
Universal Differential Equations: The Universal Differential Equations paradigm represents an ad- vancement in scientific machine learning, structurally integrating mechanistie models with data-driven com- ponents. Introduced by Rackauckas et al. (2D, UDEs parameterize unknown dynamical terms using neural networks, enabling simultaneous leverage of domain knowledge and data-driven flexibility, This extends several key lincages: Neural ODEs (5), Physics-Informed Neural Networks (22), and symbolic regression for interpretable dynamics (7). While previous UDE applications have focused on physical and biological systems, our work explores their value for financial forecasting, modeling volatility dynamics as partially- specified differential equations with learned market microstructure effects.
3 Method
We formulate financial volatility forecasting as a multi-step prediction problem. Let {a)]_0 denote the sequence of log-volatilities, where 2; In(or) and o, represents the latent daily volatility. The objective is to predict future log- volatilities 444h at fixed horizon h, leveraging past observed values within look-back window of length K.
Log-volatility is the standard target for volatility modeling. acts as a variance-stabblizing transforma- tion: the original volatility process o, exhibits hetcroskodasticity, while x, In In(a,) converts multiplicative dynamics into additive noise with more stable variance, simplifying modeling assumptions across methods.
Volatility is estimated using the Garman-Klas, (GK) range-based estimator on daily prices. For each day t, the GK realized variance is
dak, (In #;)² (2 (In ):)²
(1)
where Ot, L,, Ct are the open, high, low, and close. We take or Vošk. and use x, In(a,) for modeling. The GK estimator is approximately 7.4 times (10 more efficient than the close-to-close squared returns volatility estimator for daily data while remaining robust to drift.
3.1 Pure Physics Model: ODE with Mean Reversion
Considering the log-volatility x2 E R, the pure physics model posits the contimuous time dynamics governed by the Ordinary Differential Equation:
d(x)/((t) = n(0 - x))
(2)
with parameters x representing the mean reversion speed and 0 E IK the long-term equilibrium. This lincar vector field is globally Lipschitz continuous with Lipschitz constant k, which by the Pinard-Lindelöf theorem guarantees existence and uniqueness of solutions for all initial conditions. The closed-form solution
(t))= = + (ro - f)e
demonstrates exponential convergence to 6, confirming global asymptotic stability. The Jacobian matrix of the system linearised about the equilibrium point x² = 0 is J = -x < O, which ensures perturbations decay exponentially, embodying the mean reversion property fundamental to modeling volatility dynamics.
3.2 Pure Neural ODE Model: Data-Driven Dynamics
In contrast to the purely parametric physics model, the pure neural ODE represents the log-volatility evo-
lution via learnable continuous vector field parameterized by neural network N NN₂- R + IR,
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d(x)/d(() NNo(x))
(4)
where & denotes network parameters. Leveraging the Universal Approximation Theorem, this neural net- work, with sulficient depth and non-linear smooth activations such as GELU, can approximate any continuous vector field on a compact subset of IR arbitrarily well. The local Lipschitz continuity of NN₂ ensures, via classical ODE theory, local existence and uniqueness of solutions. However, neural vector fields impose in- herent structural guarantees like mean reversion boundedness, potentially leading to unstable trajectories or divergence, which may affect forecasting performance and interpretability.
3.3 Universal Differential Equation Model: Structural Coupling via Neural Modulation
The universal differential equation framework allows us to integrate prior domain knowledge by coupling neural modulation with mean reversion, one of the strongest studied effects in volatility:
dr = Kea (fen(()) - x)
(5)
dt
Rather than learning the dynamics function entirely from data (as in Vanilla Neural ODE) or fixing param- eters rigidly (as in pure ODE), we use small neural networks to modulate the mechanistic coefficients in state-dependent manner. Specifically, the mean-reversion rate and long-tern level are defined as:
Koff (x) = softplus(s) + exp(( (anh(NN(())) Ocft(x) + (anh(N
(6)
where NN1 and NN₂ are small MLPs (e.g., 3 hidden layers with 64 units each), and b > control the modulation strength, and 0 is small constant. This multiplicative structure ensures Kef(x) 0 (preserving dissipativity) while keeping (eH(1) bounded, thereby maintaining mean-reverling stability. Critically, the neural component modulates coefficients within the mechanistic structure rather than adding separate unconstrained drift term, which would allow the network to learn dynamics that overturn the mean-reverting property.
Incremental model composition. We first enforce physics-parameci constraints (K+ via softplus and bounded fer shifts), which forces a mean-reverting representation and prevents runaway dynamics. Building this base, we progressively incorporate additional stylized facts through compositional stacking: volatility clustering (via lagged squared returns) and jump processes (via learned jump rate and magni- tude). This progressive design enables systematic ablation of cach component's contribution to forccasting performance. Table presents the model variants evaluated in our experiments.
Integrator choice: Because the volatility data are uniformly sampled at daily frequency, we employ a fixed-slep fourth-order Runge Kutta (RK4) integrator with step size h 4=1 day for all models. This choice avoids unnccessary adaptive stepping overhead while maintaining high-order accuracy and deterministic reproducibility across thousands of rolling windows. Although adaptive solvers are beneficial for irregu- larly sampled data, setting benefits more from the contimouus-time formulation's structural properties: learning a smooth vector field that governs instantaneous rates of change rather than discrete one-step tran- sitions. This continuous representation reduces discretization bias and promotes trajectory-level stability during multi-step rollouts, even under regular sampling.
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Table 1: UDE model variants with progressive incorporation of stylized facts. Ablation studies (Section 4.3
show monotonic improvements from compositional stacking.
Variant
Dynamics Equation
UDE-Mean Reversion
dt Ket(2) (Betr (2) - x))
UDE- Volatiltyy Clustering
k.n(2) (z) (0mr( x) + 7+ r2-1
where 7+ softplus(7) +
dr
UDE-Jumps
K.t(x) (x) (fm)((x) x) + ((x) [(x)
where ((x) = (oosspouussNNN(( + J(z) tanh((VN
dr Kx((x) (fe - -x) ++ r ((x) ((x)
UDE-MRVCJ
4
Experiments
We evaluate our UDE-based volatility forecasting framework using rigorous rolling walk-forward validation across diverse financial assets. Our experimental design tests model robustness across varying market dy- namics, volatility regimes, and stress periods spanning nearly two decades.
4.1 Datasets and Evaluation Protocol
We evaluate on 10 financial indices spanning major global markets and asset classes: developed and emerging market equities, government bonds, small/large cap indices, and cryptocurrency This diverse collection exhibits strong temporal dependencies, conditional heteroskedastiiitty, and stress periods, including the 2008 financial crisis, 2020 COVID crash, and 2022 market turbulence. Daily prices are obtained from Yahoo Finance, and volatility is estimated using the Garman Klass range-based estimator. We model lolavolaty z, In(o;) as described in Section 3 Complete details and tickers are provided in Appendix A.3
Asset Classes:
Rolling walk-forward validation: We use strictly non-overlapping rolling windows to ensure realistic out-of-sample evaluation. Each window consists of sequential train, validation, and test splits with no data leakage. Windows are rolled forward through the entire sample period (Aug 2005 Aug 2025), naturally traversing multiple market regimes and crisis periods. This protocol prevents look-ahead bias and provides conservative performance estimates. Full details window sizes, stride length, and split configuration provided in Appendix A.4
Basclines: We compare against statistical models (Random Walk, EWMA, GARCH-family), deep learning approaches (LSTM, GRU), foundation models (TimesFM zero-shot and few-shot), pure mechanistic baselines (ODE-MR), and pure neural ODE baselines (Vanilla Neural ODE). This comprehensive set. isolates the contribution hybrid structural constraints.
4.2 Results
We present the performance evaluation of our UDE-based architecture across multiple forecasting horizons, comparing against traditional econometric models, deep learning baselines, and foundation models. Table 2 shows aggregated results across horizons 1, 5, 10, and 20 steps ahead.

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Table 2: Normalized RMSE (mean + 95% CI). Normalized by standard deviation of observed volatility. Scale-independent metric, lower is better.
Model
Type
NRMSE
NRMSE
NRMSE 10
NRMSE
Random Walk
Baseline
0.770 + 0.063
0.841 1 0.080
0.882
0.084
0.911 - 0.087
EWMA
Statistical
1.381 + 0.220
1.422 1 0.223
1.498 0.232
1.602 0.245
(I)
Econometric
1.384 = 0.222
1.494 + 0.234
1.606
0.247
1.852 0.277
FIGARCH((.))
Econometric
1.430 = 0.235
1.548 1 0.249
1.677 + 0.267
1.985 = 0.314
LSTM GRU
Sequence
0.811 + 0.052
0.825 + 0.048
0.855 + 0.049
0.965 = 0.047
Sequence
0.803 = 0.048
0.817 + 0.046
0.818 0.048
0.906 = 0.046
MertonJD
Jump sion
Diffu-
0.834 + 0.083
0.892 + 0.115
0.929 + 0.165
0.964 = 0.196
TimesFM
Transformer
0.704 + 0.055
0.782 + 0.051
0.794 0.053
0.819 = 0.050
TimesFM few-shot
Transformer
0.701 + 0.054
0.780 + 0.050
0.791 +: 0.052
0.814 = 0.049
Vanilla Neural ODE
Neural ODE
0.735 = 0.056
0.812 + 0.073
0.949 + 0.228
0.998 = 0.090
ODE-MR
Physics
0.758 1 0.080
0.985 1 0.070
0.997 0.069
1.020 + 0.085
UDE-MR
UDE
0.662 = 0.056
0.740 + 0.050
0.753 + 0.050
0.780 = 0.050
UDE-MRVC
UDE
0.660 = 0.056
0.738 0.051
0.749 0.052
0.774 = 0.052
URV-M
UDE
0.658 + 0.052
0.735 + 0.048
0.743 + 0.049
0.759 + 0.048
Multi-Horizon Forecasting Performance Across Models
1.05
8
1.00
7
2
0.95
1. Random
2. MertonjD
0.90
4
3. LSTM
4. GRU
NRMSE
5. TimesFM
6. TimesFM-FewShen
7. Neural ODE
5
8.
9. UDE-MR
9
10. UDE-MRVC
10
11. UDE DRVCI
0.75
11
0.70
0.65 1=1
h=5
h=10 Forecast Horizon (h)
h=20
Figure 1: Multi-horizon forecasting accuracy (NRMSE; lower is better) at horizons h E (1,5, 10, 20) for base- line (Random Walk), sequence models (LSTM, GRU), transformer models (TimesFM, F M-FFewhho)), and Neural-ODE/Unt /UDE variants. The UDE-MRVCJ model (incorporating mean reversion, volatility cluster- ing, and jumps) achieves the best performance across all horizons. Curves show results over rolling evaluation windows; lower lines indicate more accurate forecasts.
We evaluate significance using the Diebold-Mariano (DM) test across rolling windows for each asset. Because the selected assets span distinct classes (equity indices, emerging-market indices, and crypto) with low cross-

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Table 3: Whebold-Mariamt significance across assets (combined p-values, Fisher). Lower indicates model 2
significantly outperforms model
Pair
h
10
Vanilla Neural ODE + UDE-MR
3.1e-4
1.8e-5
2.2e-6
8.5e-7
UDE-MR UDE-MRVC
0.0170
0.0857
0.0709
0.0386
UDE-MIVVC - UDE-MRVCJ
0.0412
0.0013
0.0034
0.0189
correlations in realized volatility, we treat test outcomes approximately independent. Combined p-values are therefore computed using Fisher's method.
4.2.1 Performance Analysis
UDEs combine accuracy and stability. UDE variants slightly outperform the foundation model (TimesFM: 0.70; UDE-MRVCJ: 0.66) at short horizons. Across longer horizons, UDEs remain stable (0.66 0.76 from h = 1 to h 20), whereas purely neural models degrade (Vanilla Neural ODE: 0.74 1.00) and rigid mechanistic models stay suboptimal (ODE-MR: 0.76 + 1.02).
Compositional stacking yields monotonic improvements. Table g shows statistically significant improvements at each composition step. Adding mean-reversion modulation to Vanilla Neural ODE provides the largest gain (DM p < 3.1e-4 at h = 1, < 8.5e-7 at = 20). Volatility clustering (UDE-MRVC) and jumps (UDE-MRVCJ)) provide incremental but consistent improvements, particularly at longer horizons where structural constraints prevent catastrophic degradation.
4.2.2 Computational Efficiency
Table 4: The efficiency table highlight that UDE variants are orders of magneitude smaller and have a smaller footprint than large neural and foundation baselines while preserving long- morizon stability.
Model
Params
Memory (MB)
Train (s)
Infer (s)
Total (s)
Epoch (s)
Random Walk
0
0.00
N/A
0.00
0.00
0.000
EWMA
0.05
N/A
0.00
0.00
0.000
G1RC
0.10
0.36
0.00
0.36
0.000
FIGARCH(1,1)
0.10
1.11
0.00
1.11
0.000
Merton JD
0.15
0.95
0.02
0.97
0.002
LSTM
50.5K
0.19
0.19
0.19
0.68
0.010
GRU
37.9K
0.14
0.46
0.18
0.64
0.009
ODE-MR
0.00
0.50
0.02
0.52
0.005
Vanilla. Neural ODE
4.4K
0.02
1.02
0.05
1.07
0.010
UDE-MR
4.4K
0.02
1.32
0.09
1.42
0.013
URV-
4.4K
0.02
1.46
0.10
1.56
0.014
UDE-MRVCJ
4.4K
0.03
1.54
0.11
1.65
0.016
FimesFM
200.0M
800.00
N/A
25.52
25.52
0.000
FimesFM A--Fehhhot
200.0M
800.00
N/A
25.25
25.25
0.000
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4.3 Ablation Study
Table 5: Ablation study of model components for multi-step volatility forecasting. R measures explained variance; sMAPE measures scale-independent forecast accuracy (lower is better).
Model
Structure
h =
h
20
R²
sMAPE
R²
sMAPE
ODE-MR
Pure Mechanistic
0.421
7.91
-0.198
11.58
Vanilla Neural ODE
Pure Neural
0.456
7.50
-0.136
10.46
UDE-MR
+ Mean Reversion
0.518
7.17
0.150
9.37
UDE-MRVC
+ Vol. Clustering
0.522
7.14
0.151
9.36
UDE-MRVCJ
+ Jumps
0.535
7.03
0.160
9.11
4.4 Interpretability via Parameter Evolution (S&P 500)
We track how the learned parameters our -RD-MJVCJ evolve across rolling windows for the S&PP 500. This reveals where the model adapts and where remains invariant. First, the mean-reversion speed s(x)) and the long run level ((x) co-move with regime shifts: decreases during prolonged drawdowns (slower pullback loward equilibrium) and 0 rises when background volatility is persistently elevated. Second, the jump mechanism is event sensitive: the estimated jump rate A(z) exhibits pronounced spikes around market stress (c.g., macro announcements, crisis windows). while the jump magnitude concentrates within those episodes, indicating a parsimonious "rare event" channel rather than persistent overfitting. Third, the volatiltyycclustrring strength (7) varies smoothly and remains bounded, capturing short-horizon persistence without destabilizing the core dynamics. Finally, a stability summary and correlation heatmap show that core parameters (x. remain stable across windows and are only weakly confounded with camp/clusterngg terms, supporting identifiability and interpretability. Together, these diagnostics let explain forecast changes (e.g.. "volatility increased because mean reversion slowed and jump intensity spiked") rather than treat predictions opaque.
Mcan Reversion Speed (k) Evolution
Long-term Mean (0) Evolution
CDE-MR
0D6-MR
0.60
UDE-MRIVCJ
UnE-AI UDE-MRVVC
-4.425
-4.450
0.55
space)
-4.500
0.50
-4.525
0.45
-4.550
-4.575
0.40
1000
2000
3000
4000
5000
6000
7000
1000
2000
3000
4000
5000
6000
7000
Start Index
Window
(a) Mean-reversion speed <(œ)): Controls how quickly volatility returns to equilibrium. Higher values during stress indicate faster normalization.
(b) Long-run level ((x) across windows (S&P 500). Equilibrium volatility target. Increases during crises (el- evated baseline risk), decreases during calm periods.
Figure Learned mean-reversion dynamics across market regimes. Pure mechanistic baseline uses fixed parameters and oscillates aggressively in response regime shifts, while UDE, variants' neural components absorb these corrections, leading to more stable and meaningful parameters.
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Parameter Correlation (VoiCI+jumps)
1.00
0.75
0.50
VolCi+jumps: All Parameters Combined
0.35
0.25
0.30
0.00
0.34
0.25
lgg Dump 5000
0.50
0.20
-0.25
leg(A)
(Meen
0.15
0.48
-0.50
0.28
0.10
ssize
-0.7
0.46
0.05
0.00
logrA)
size
-1.00
0.44
0.24
1000
2000
3000
4000
5000
6000
7000
Window Start Index
(a) Parameter correlation: weak confounding across channels (S&P 500).
(b) Composile view: clustering jumps aligument with stress (S&P 500).
Figure 3: Identifiability: mean reversion parameters are largely orthogonal to jump/clustering terms.
Jump Rate Evolution (avg A=1.1331 jumps/step)
Jump Magnitude Evolution (avg=-0.214)
0.35
log(A)
0.4
Jump Size
No jump
0.30
0.25
Jamp (logwol))
0.20
log(A
0.15
-0.2
0.05
-0.4
0.00
-0.6
2000
3000
4000
3000 4000
5000
Window Start Index
Window Start Index
(a) Jump rate lambda(() Average jumps per step. Spikes during stress periods indicate the model learns when jump-like dynamics are necessary.
(b) Jump magnitude J(x): Size of jump effect when acti- vated. Concentrated during tail episodes, near-zero oth- erwise.
Figure 4: Jump dynamics activation during market stress (S&P 500). Our smooth, continuous jump formu- lation approximates the aggregate effect of discontinuous jumps through deterministic modulation.
5 Limitations and Future Work
Point forecasts vs probabilistic predictions. Our framework produces deterministic point forecasts rather than distributional predictions with uncertainty quantification. While sulficient for demonstrating Multi-step stability, risk management applications require value-at-riets estimation or full predictive distri- butions. Neural Stochastic Differential Equations (Neural SDEs) offer a natural extension by adding learned diffusion terms:
dx/dt Kedt ken(w) (x))(()(00(())) 2d] (eWW
(7)
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Parameter Stability Across Windows
Volatility Clustering Strength (y) Evolution
0.05
0.35
Allmm M WannMM
0.04
0.30
Standard Deviation
happ
0.03
Learne
0.20
jumpss Y y=0 clustering)
0.02
0.15
0.10
0.01
0.00
* (Conat)
a (Const)
(Hybrid)
(Hybrid)
(VViCI++jp)
a (oiCH+)mm)
0
1000
2000
3000
4000
5000
6000
7000
Window Start Index
(a) Clustering strength y: smooth, bounded variation
(b) Stability summary: core parameters vary modestly. Consistent values across windows indicate the model learns general representations rather than overfitting.
(S&P 500).
Figure 5: Parameter stability and clustering dynamics (S&P 500). Volatility clustering strength varies smoothly without destabilizing core dynamics. Standard deviations across rolling windows confirm core mechanistic parameters remain stable while jump/chstering terms show comparable consistency.
where (z)(()) models stale-dependent volatility and W; represents Brownian motion. However, Neural SDEs remain notoriously difficult train: adjoint methods require solving backward through stochastic processes with numerical instability (15). In contrast, the diffusion coefficient design requires careful constraints to prevent explosive trajectories (20). Recent advances in stable Neural SDE classes and signature-based training (14) suggest promising directions, but adapting these to compositional UDE frameworks remains open work.
Limited exogenous features: We use only lagged squared returns as exogenous input. Incorporating macroeconomic indicators, market microstructure features, or cross-asset correlations could improve regime detection and forecasting accuracy. In practice, horizon volatility dynamics often depend on macroe- conomic indicators, option-implied volatility, and cross-asset information. Integrating these features could enhance regime detection and longerange accuracy, though it risks diluting the structural clarity and parsi- mony that make UDEs interpretable. A key open question is how to incorporate exogenous signals without sacrificing mechanistic transparency.
Domain generalization: While financial volatility provides an excellent testbed due to documented styl- ized facts and accessible data, demonstrating the framework's offectiveness across diverse domains would strengthen claims of general methodological contribution. Each domain would require careful physics term design and validation against domain-specift. baselines.
Despite these limitations, our core results remain robust: UDE variants with inductive biases achieve stable multi-step forecasting with dramatic parameter efficiency and interpretable dynamics. We explicitly note that institutional-grade data would be required for deployment. in real-world trading or risk management systems.
6 Conclusion
We demonstrated that Universal Differential Equations (UDEs) provide an effective framework for multi- step time series forecasting, achieving superior accuracy and long-horizon stability across diverse financial
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regimes, including crises and structural breaks. Our compositional UDE variants consistently outperform foundation, econometrie, and deep learning baselines while using orders of magnitude fewer parameters.
UDEs and neural differential equations more broadly are particularly well suited to forecasting because they model temporal evolution as a continuous flow rather than a fixed-step recurrence. Even with regularly sampled data, this formulation enforces trajectory-level smoothness and reduces discretization bias, the main driver of error accumulation in recurrent architectures. By learning instantaneous rates of change instead of discrete transitions, UDEs maintain stable dynamics over extended horizons.
More fundamentally, UDEs offer principled bridge between mechanistic and data-driven modeling. By embedding domain structure, mean reversion, volatility clustering, and jump effects within a differential equation while allowing neural modulation of key coefficients, they preserve stabilizing inductive bi- ases that confine the learned dynamics to economically plausible regimes. Our compositional framework showed that each successive incorporation of domain-informed structure yiclds monotonic, statistically significant gains, demonstrating that carefully chosen inductive biases can substitute for brute-force scale.
This combination of continuous-time modeling and structured knowledge integration explains why compact UDE models can rival or surpass massive foundation models. Structural bias, when aligned with the data- generating process, not only enhances efficiency and interpretability but also delivers robust multi-step forecasts that remain coherent across regimes, an essential property for modeling inherently unstable systems like financial volatility.
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References
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[8] Abhimanyu Das, Weihao Kong, Rajat Sen, and Yichen Zhou. A decoder-only foundation model for time-series forecasting. Tn International Conference on Machine Learning (TCML), 2024. arXiv preprint 1rXiv:2310.1068..
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[18] Sohcil Almasi Monfared and David Enke. Volatility forocasting using a hybrid gir-garch neural network model. Procedia Computer Science, 36:246-255 2014. Complex Adaptive Systems Philadelphia, PA November 3-5, 2014.
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[19] Yuqi Nie, Yulin He, Qingsong Wen, Liang Sun, Rong Jin, and Ziqing Ma. A time series is worth 64 words: Long-term forecasting with transformers. In International Conference on Learning Represeniations (ICLR), 2023.
[20] YongKyung Oh, Dong- Young Lim, and Sungil Kim. Stable neural stochastic differential equations in analyzing irregular time scries data, 2025.
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[22] Maziar Raissi, Paris Perdikaris, and George E. Karniadakis. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. Journal of Computational Physics, 378:686-707, 2019.
[23] Souhaib Ben Taieb and Rob J. Hyndman. Recursive and direct multi-step forecasting: The best of both worlds. In Proc. International Conference on Neural Information Processing, pages 30 40, 2012.
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Oral Presentation.
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[26] Xiang Xie, Ajith Kumar Parlikad, and Ramprakash Srinivasan Puri. A neural ordinary differential equations based approach for demand forecasting within power grid digital twins. In 2019 IEEE In- ternational Conference Communicntions, Control, and Computing Technologies for Smart Grids (SmartCridCommm)) pages 1 1-6, 2019.
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A Appendix
A.1 Lipschitz Continuity for UDE-MRVCJ
We consider the one-dimensional ODE on log-volatility
dr
= f(2;1(-1) = (x) (fef( - x) +
+ J(x)
dt
vol. clustering
experted jumps
where
Keft (I) Ko exp(s tanh(k(x)))
(L + tanh(O(a)),
A(x) = softplus(((1)) +E,
J(z) = tanh (j(z)).
Here Ko softplus(x) + > 0 is constant, > 0 is the modulation scale, b > 0 is the O-shift bound and > is a fixed numerical slack. The maps k, 0, L,j R - R are the scalar outputs of two MLPs with GELU activations (onc for (k, 0) and one for {e,j)). The clustering input r²- exogenous and treated as constant in at each step. We assume the MLPs are globally Lipschitz with constants Luase (for k. 1) and Zijupp (for (,j). For GELU networks this holds with
LMPP &
(II
[W.ll2) (1.129) # #activations
We write K [M,, for a compact domain of interest and use || for the Euclidean norm R.
Auxiliary bounds:
(i) Since |tanh' S 1 and (softplus) = =oe 10,
(l))) s [g [(sottplus o g)'l 1g'1,
Vg R R. Lx = KD Libases
(ii) For r E K, define
Kmax Kg
La Liaas.
Then [=t((z)] S Kmax and [x)r S Lus also 10'r (x)] S La and (0tr(x)) S 10 (iii) For the jump factors, [[x)] < 1 and
[N(x))
Ljump:
[J((x)) & Tjump-
Let
A(M)= = sup A(x) < log + |8(0)) + Ljump M + =ek
(using softplus(t) < log(1 + eld) < log + | and |((x)) < |((0)] + Iyumpp (z:)).
Theorem A.1 (Local Lipschitz on compact sets). On any compact = [-M, M,, the vector field f(;rf-1)
is Lipschitz continuous in x with constant
L₁(M) S LA (M + 101 + b) + Kimx (Lo + 1) + Ljump (1 + A(M))
mean reversion term
expected jump
The erogenous clustering term 7 does not affect the Lipschitz constant x.
Proof. Write f(x) = ((())(()++++g(() where g = Kefff Octt I, c 74rī- (constant x)), and q XJ. For any 31, x2 K,
1g1h1 g2h2l I 1g1 92 1h1 + 19el l11 h2l-
Using [M(x)) < (x)) + |a|| S |0| + b M and |/1 h22 S (To + 1)|=1 x22. plus 1g1 g2l T. - x22
and [g2l < Kmax: we obtain the first bracket. For q = XJ,
[ - < AA A211 + Dal 1 - J2l < (Ljump 1 +AMM) Ljump) |21 22/-
Summing the bounds yields the stated (M).
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Theorem A.2 (Linear growth and global existence). There erist constonts Co(M), C1 2 0 such that |s x Co(M) + C1 |zb
One admissible choice is
C
Nmax + jjumps
Co(M) Kimax (10) + b) + h4 1 log 2 (e(0)) + +,,
using X(x)= S log + | + Ljumplat lat + Consequenlly, by Picard Lindelöf and Grönwall,, the IVP admils a unique solution globally in time.
A.2
Convergence of RK4 for Our Constrained Neural Reversin with Clustering and Jumps
We study our model
=(t) = f(x(1), = (e)( (0cm(x) - x)
+
7 u()) vol. clustoring
+ X(x) J
mean reversion
expected jumps
with
Keft (er) No exp(s tanh(k(()))
(x)(x) 0 + tanh(O(x)), J(x) = tanh()(x)),
X(x) = softplus(((()) ((r) +
where k; 0,, J are scalar outputs MLPs with exact GELU activations, and u(t) = - is bounded exoge- nous regressor. In our solver, we hold u(t) piecewise constant on each RK4 step (matching our experimental procedure).
Let [) T] be fixed and K C R be compact set that contains the true trajectory and all RK4 stage values (for & small enough).
Hypotheses we verify. - -HH) Uniform Lipschitz in x on K: by our Lipschitz result in Appendix A.1 there exists L such that [f(x1, - (x2, u)l S L |2] - x2l for æ1, K and admissible u. -(H) Smoothness in x on K: exp, tanh, softplus and exact GELU are C compositions with affine maps preserve C u) C (K) uniformly in u. Hence 8Thh f(; u) are bounded on K for m 0,...: 4. (H3) Stage
confinement: since f is bounded on K, there exists hyg > 0 such that for 0 < h s ho all RK4 stages remain in K.
We apply classical RK4 with step h > O, freezing u(t) = un on [tns fnt1), bn = nh:
1 = f(En, un), = = f(r + un) h3 = |æn + 2/2 k2 un),
k4 = fzz + hks, un).
Int1 In + h (ki + 2k 2kg + ka))
Lemma A.3 (Local truncation error (order 5)). There erists Cloc = CloeK (K, such that, for all x E K and admissible (frozen) u,
|| u) (n(a, uill s Cloc h² where 9h is the exact time-h flow of x' f(x, u) and 4 is one RK4 step.
Proof. By (H2), f(.,u) E (K) uniformly in u. The give a uniform O(15) remainder. (H3) guarantees stage evaluations stay in K where the bounds apply. ]
Butcher expansion and RK4 order conditions
Lemma A.4 (One-step Lipschitz stability). There erist = =c cI(KK and h > 0 such that, for 0 < h S h and all x, y E K and admissible u,
[I u) - Cn(y, w)ll S (1 + c1 ||0- ---l- vll-
Proof. Write (2, u) = x + h (x, u) with stago-averaged slope Vn- By (H1)-(H2), (.(.u) is Lipschitz
in x with constant + O(h) uniformly on K. Hence [[ u) (,, a)ll < (1 h(L + C'))l² yll for some C", proving the claim.
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Theorem A.5 (Global (It)) error for our model). Under (H1)-(H3) there crists C C(K, f), independent of h and n, such that
max USHEN
[k((t)) - znll S C CeLTTe 1t,
N = [T/h].
Proof. On Inn 1n+1) with frozen un, the exact solution satisfies s(In+1) = yn (p((((()) un), , so
ex+1 == x(fn+1) )- - 2a+1 = [2 un) 2 (2ns un)] + Tn+l.
| S Chart.
By Lemma A.4
|eatill || I (1 + celllen + Cloch.. Discrete Grönwall yields ||eall - Chatthh & C
since e 2 L and ep = 0.
Why the hypotheses hold for our architecture. Mean reversion term: Ketf and feff are C compo- sitions of exp, tanh and affine maps; on K they and their first four derivatives are bounded. Our Lipschitz constant L K was established Appendix A.1] Clustering term: 74 u(t) is constant æ and bounded in I, it does not affect Lipschitz in nor the order of the method. Jump term: À softplus l and J tanh oj are C their products and derivatives bounded on K. Stage confinement: since is bounded on K, taking h S ho(K, ensures stage values remain in K.
A.3 Datasets
We evaluate the following 10 indices spanning major global markets and asset classes:
Developed Market Equities: S&P 500 ("GSPC), NASDAQ ("IXIC), Dow Jones (*DJI)
Emerging Market Equities: NIFTY50 ("NSEI)
Small/Large Cap: Russell 1000 ("RUI), Russell 2000 ("RUT)
Government Bonds: iShares Core U.S. Aggregate Bond ETF (AGG)
Cryptocurrency: Bitcoin (BTC-USD), Ethereum (ETH-USD)
International: FTSE 100 ("FTSE)
Daily OHLC prices are obtained via yfinance. Realized volatility is estimated using the Garman -lassn-KKasss estimator (Fq. D and model log-volatility 2; In(o.). Evaluation period: August 2005 to August 2025.
A.4 Evaluation Protocol
Rolling window configuration.
We the following strictly temporal configuration:
Train: 63 trading days (=33 months)
Validation: 20 days
Test: 30 days
Stride: 30 days between consocutive window start dates
We selected the training window length via small hyperparameter sweep over candidate lookback periods 7, 21, 63, 126, 252 trading days and evaluated normalized MSE held-out rolling windows across all assets. The 63-day window produced the lowest aggregate NRMSE across assets. Intuitively, longer windows (126, 252) often span multiple volatility regimes, causing Iearned modulators to average over heterogencous dynamics and thereby reduce predictive utility; visually, parameter trajectories est estimated from 126/252 22y fits show piecewise regime structure that a single parameter set cannot capture reliably. A short window (7, 21) lacks sufficient information estimate even the compact neural modulators stably. Thus, the 63-day choice represents an empirical tradeoff between enough data fit modulators and narrowness of the regime assumed.
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Walk-forward guarantee of no data leakage. The fork--orward protocol ensures strictly out-of-sample testing by moving forward in time only. For each window, we:
1. Train a model on the training segment
Validate on the validation segment
3. Make predictions on the test segment
Roll forward by stride days and repeat
Critically, we never revisit past test periods. Once predictions are made for a test segment, that window is complete. Even though future training windows may include data that was previously used as test data, we never re-predict those periods with the benefit of future information the predictions have already been made and recorded using only past data. This is the defining property of walk-forwart validation: we only predict forward in time, never backward.
For train size of 63 days, validation size of 20 days, test size of 30 days, and stride of 30 days:
Window Train on days 1-63, Val on days 64-83, Test on days 84-113
Window 2:
Train on days 31-93, Val on days 94-113, Test on days 114-143
Window 3: Train days 61-123, Val on days 124-143, Test days 144-173
By Window 3, days 84-113 (Window Lest set) now appear in the training data. However, predictions for days 84-113 were already made in Window using only data up day 83. We do re- re-predict them. This ensures that every prediction uses only information available at the time the prediction would have been made in real-world deployment.
Forccast horizons and autoregressive rollout. We evaluate at h E {1,5, 10, 20} days ahcad. For = 20, each 30-day lest window yields 10 independent predictions (days 1-20, 2-21, ... 10-29 of the test period).
All predictions are made using strictly autorcgressive rollout with no future data leakage. When predicting al time l for horizon h, the model uses only information available up time Critically:
For multi-step predictions (h > 1), the model recursively feeds its own predictions forward
No ground-truth values from t + 1 to + h - are provided during the rollout
Exogenous variables (c.g., squared returns r²-1) 1) use only strictly causal values: when predicting at lime I, we use r²-1 (available), never r² future returns
This ensures realistic evaluation: predictions are made exactly as they would be in real-world deployment, where future information is unavailable.
Example of autorcgressive rollout. To predict log-volatility at + 20 starting from time t:
1. Input: observed log-tolatilities up to f, exogenous features up to t - (e.g., rī-1)
Prodict d using the ODE integrated from f++1
3. Predict 4.-2 using L.+1 as imput (not ground truth 2x41)
4. Continue rolling forward: 2443 from Set2, .., $4420 from 1
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5. Exogenous features at each step only past information (e.g., at step t + k predicted f + derived from fx+4-1)
This protocol tests the model's ability maintain stability under compounding error, the key challenge in multi-step forecasting.
A.5 Training Configuration
Unless stated otherwise, we use:
Seeds: 42, 43, 44 (results aggregated across all 3 seeds and 10 assets)
Horizon in loss: Nssteps = 20
Batch size: 50
Networks: Small MLP modulators with GELU activations
Hidden sizes tested: [64,64], [64,64,64] [64,,4,664,44],, [16.16].[32.3]]
Selectod: [64,64] (no further improvement with deeper architectures)
Optimizer: Adam with learning rate 1 10-3
Epochs: 50 (with early stopping on validation loss)
ODE Integrator: RK4 with fixed step size
Constraints: Softplus for mean-reversion rate (K4 > 0), bounded shifts for long-rum level (6ea)
A.6 Compute Environment
Experiments were run on Linux/macOS with Python 3.10-.111 GPU was optional; CPU results were consistent modulo runtime.
Core libraries:
PyTorrc
Neuromancer for ODE integration, systems, and losses
NumPy 2 1.26, SciPy 2 1.11, Pandas 2 2.0, Matplotlib 2 3.8
A.7 Evaluation Metrics
NRMSE.
Normalized Root Mean Squared Error, normalized by the standard deviation of observed volatil- ity (scale-independent metric, lower is better).
Statistical significance. We report Diebold Mariano tests (Fisher combined across assets) to compare model pairs using two-sided squared-error differentials. No multiple-comparison correction is applied. Com- bined p-values are reported in Table
Diebold-Mariano
A.8 Econometric Baselines
MARCH-famill models (GARCH, FIGARCH) were estimated using the arch Python library. Models were refit. in each rolling window via maximum likelihood. Convergence was verified; the best-performing config- uration after convergence reported.
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A.9 Per-Asset Performance Metrics with Regime Breakdown
This section presents detailed per-assel performance metrics across all forecast horizons (h E {1. 5, 10, 20)). broken down by volatility regime. For each asset-model combination, we report 12 metrics: MSE MAE, QLIKE, and sMAPE for (1) aggregated data, (2) low volatility regime, and (3) high volatility regime. All values are rounded to decimal places. Bold values indicate the best-performing model for each metric-regime combination within each asset.
Regime definition: Per asset, classify volatility using rolling Z-score of realized volatility computed centered rolling window (63 steps). Low-vol if z -0.5; high-vol if z +0.5; observations in between are not used for regime slices.
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Table 6: Short Horizon 1) Performance: Crypto Bonds
Agreggaedd
High
*M
MBE
QL
Biscoin
RW
0.37
0.45
1.07
12
0.43
0.33
0..58
122-70
0.37
2.49
ODE-MR
0.35
0.45
1.35
12 95
0.44
0.55
0.27
13.42
0.84
0.88
4.08
23. 28.08
Vanilla Neural ODR
0.35
0-47
0.58
12 16
0.42
0.32
0.55
12:48
D-46
0.54
2.31
15:81
UDE-MM
0.37
D-41
0.83
10 85
0-43
0.36
0-40
7.42
D-45
0.52
0:42
17:17
UDE- MUVC
0.33
0-40
D-80
10 89
D-42
0.45
0-40
7.42
D-47
0.51
0-40
17.17
UDE-MRVCJ
0.36
0.39
0.70
10.68
0.43
0.46
0.39
10
4G
0-50
0.88
16.96
GARCH
0.80
0.78
0.02
1:76
1.23
1.59
32
0.56
04
12 12.93
0.79
0-85
1.58
1.48
0.55
12.58
FIGARCH
0.82
1.68
1.49
0.84
MertionI
0.57
1.78
1.24
0.94
TinccaM
0.35
0.45
3.65
16.95
TimessP
0.34
0.45
0.95
2.54
16.96
Ethereum
RW
0:31
86
16.85
OnR- MIt
0.44
0.55
0.37
0.27
13.34
1.06
28.37
Vanilla Neural ODR
0.29
0-45
12
1.77
16:34
UDE-MR
0.31
0.81
0.71
10 80
0.34
0.28
0.3G
5.80
0.44
0.44
0.81
17:49
MRVC
0.31
0.81
0.69
10 87
0.34
0.34
0.34
10:23
0.44
0.44
0.31
17-40
UDE-MRVCJ
0.30
0.30
D-6S
10.52
0.34
0.28
0.33
6.74
D-45
0-455
0.51
17.22
GARCH
0.75
0.70
0.75
21 08
1.44
1.11
1.35
31.80
0-42
0.54
0.48
19.05
EWMA
0.85
0.68
0.72
19.61
1.28
1.06
1.26
20
0.41
0.53
0.46
19.26
PIGARCH
0.64
0.65
0-71
1:28
1.04
26
29
0-52
0-44
MertonII
0.89
0.69
D-75
1-41
1.12
1.38
91.52
0.25
TimccuF3
ACC
RWV
7.96
ODE-ME
0.26
0.44
7.50
Vanilla Neural ODE
0.24
0.39
0.43
0.27
0.41
0.24
0.47
7.76
UDE-MR
0.25
0.25
0.38
0.45
0.26
0.28
0.20
8.85
0.34
0.34
0.63
7.20
UDE-MRVE
0.24
0.24
0.40
B.85
0.26
0.28
0.26
8.63
0.85
0.35
0.47
6.85
UDV-MRVCJ
0.24
0-24
D-43
6.30
0.26
0.26
0.31
3.56
D-35
0.35
0.35
7.00
GARCH
1:27
1.00
1.16
17
2.10
1.37
1-54
22.75
D-18
0.59
0.64
10:71
EWMA
1.18
0.96
1.00
16 16.42
1.98
1.32
1.58
21.79
0.48
0.57
0.59
10.41
FIG ARCH
1.13
0.93
1.05
15.87
1.89
1.30
1.51
21.42
0:41
0.52
0.56
0.57
MertonJD
1.35
1.05
1.22
17.81
2:23
1.04
1.75
23.87
0.53
0.63
0.61
11.13
0.51
0.67
0.89
0.78
12.62
0.28
6.55
0.35
0.40
0.86
Times?M F.
21

---

Published in Transactions on Machine Learning Research (12/2025)
Table 7: Short Horizon (h- 1) Performance: Equities
Aggregated
High Vol
Azret
Model
MBR
MSE
MAE
xM
Equitics
DowJonee
RW
0.22
0.37
0.54
7.38
0.23
0.39
0.34
7:22
0.28
0.44
1.05
9.06
ODE-MR
0.29
0.44
0.39
8.63
0.55
0.70
0.00
13.3T
0.29
0.40
0.28
9.49
Vanilla Nevrral ODE
0:21
0.37
0.51
7:22
0.23
0.39
0.333
7:18
0:24
0-43
1.02
8.00
UDE-MR
0.22
0.22
D.38
6-5/9
0.23
0.23
8.96
0:27
0.24
0-41
5.33
UDE-
0.21
0.21
0.38
0.56
0.23
0:27
0.27
DDB-MRVCJ
0.22
0.22
0.78
0:28
0:25
0:39
EWMA
0.77
0.72
FICARCH
0.74
0.70
MertonJID>
0.85
0.77
0.59
16.36
1.16
1.47
23.
0.23
0.59
0.37
LSTM
0.47
0.54
1.96
0.40
0.5%
9.49
0.63
0.68
3.21
13.97
GRU
0.48
0.54
1.51
10.78
0.55
0.B8
11:24
0.63
0.68
3.21
13.97
TimesPM
0.27
0.40
0.7G
T.04
0.34
0:48
O. 44
9.09
0.36
0.48
1.70
9.92
Ti PS
0.24
0.39
0.64
T:55
0.32
0 4-47
D. 42
5.55
0.31
0.44
1:37
9.11
PTSR
RW
0.23
0.38
0.66
T-56
0.24
0-40
0:34
7.56
0.37
0.48
6.66
10.11
QDR-MI
0.27
0-43
D.38
B-16
0.54
D. 69
66
13.18
0.3T
0.32
0.27
10.92
Vasilla Nearal ODE
0.28
0.38
0.68
7.45
0.24
O. 40
7.60
0.35
0.48
1.60
9.92
UDE- MR
0.28
0.30
0.41
0.58
0.24
0.24
6.95
0.36
0.36
0.3G
5.52
UDE- MRVC
0.23
0.23
0.07
0.24
0.24
14
0.36
0.36
0:37
64
UDE-MKVOL
0.23
0.23
0.39
6.44
0.24
0.24
0.37
0.26
0:37
5.10
0.58
0.02
0.00
12.57
18.4
0.19
0.36
0:35
RWM
FIGARCH
Marton3I L8TM
0.35
0.47
9.19
0.
0.76
0.50
3.68
16.03
GRU
0.35
1.00
0.61
0.70
2.75
14.08
TimePM
0.22
0.39
0.62
7.27
0.24
0.
0.94
7.02
0.12
0.57
1.69
10.02
Times FS
0.20
0.36
0.54
7.00
0.24
0.
0.95
7.64
0.34
0.47
1.86
9.74
NASDAQ
RW ODE-MR
0.25
0.38
0.61
T.63
0.28
O.
O. 84
7.08
0.32
0.46
1.81
10.16
0.29
0.38
0.52
S.50
0.48
0.00
O. GS
11.00
0.35
0.51
0.85
10.02
Vasillaa Neural ODE
0.25
0.37
0.48
T-66
0.26
0.39
D. 45
7.70
0.31
0.45
0.73
9.90
UDE-MR
0.24
0.35
0.44
7-52
0.30
0.41
D.BO
0.50
0.32
0.43
0.62
8.85
DRE-MRVC
0.22
0.22
0.39
7:22
0.23
0.28
0.58
7.0T
0.51
0.31
0.42
6.98
UDDEMRVC
0.29
0.24
D.36
7:21
0.23
0:34
7-70
0.30
0.32
D-40
7:05
GARCH
0.91
0.81
0.94
18.15
1.87
1-55
26. 17
0.24
0:41
D-41
EWMA
0.85
0.
0.88
17.23
1-57
1.17
0:25
0.38
0:39
FIGARCH
0.52
0.77
1.64
84
0.24
Merton3D
0.91
TO
0:25
L8TM
GRU
0.54
19.ST
13.03
TuscPM
0.29
0.42
0.91
10.18
10.56
Tima.PM PS
0.27
0.40
0.80
K.34
0.36
0.49
10.02
0.34
0.48
1.71
10.07
SIPTY50
RW ODE-MR
0.22
0.37
0.5T
7.30
0.33
0.25
0.41
0.35
5.10
0.52
0.67
12.79
0.32
0.51
0.22
10.34
Vanilla Neural ODE
0.21
0.36
0.54
7.18
0.24
0.39
0.35
1.ST
0.32
0.48
1.24
9.30
UDDE-RRR
0.22
0.22
0.37
6:45
0.25
0.25
D. 35
5.17
0.33
0.23
0.43
5.05
UDD- MRVC
0.22
0.22
0.36
6.11
0.25
0.25
D. D.
5..01
0.32
0.32
D-44
7.92
UDE-MRVCJ
0.21
0.21
0.36
6.31
0.25
0.25
0.32
7:73
0.32
0.32
D.41
7-61
GARCH
0.54
0.77
0.ST
16.47
1.43
1.13
1.37
22.77
0.31
0.41
0.40
5.54
EWMA
0.51
0.76
0.54
16.01
1.41
1.10
1.
23. 50
0.37
0.39
0:35
9.04
PIGARCH
0.78
0.74
0.52
10. 78
1-37
100
1333
22.
0.24
0.30
0:37
8.14
MartonJJ
0.38
0.80
0.91
17.05
1.49
25.02
0.28
0.43
0.40
0.48
0.50
9.57
0.25
0.25
0.80
3.41
Time.PM
TimaPP
Russell100
RW
ODE-MR
0.48
18.56
14.38
Vanilla Nenral ODE
0.61
0.54
3.31
10.01
0.52
9.26
1.04
0.75
9.70
14.36
UDB-MR
0.38
0.37
0.52
N.12
0.40
10.71
0.49
0.35
1.05
5.12
UDE- MRVC
0.33
0.34
0.61
7.02
0.36
O. 85
O. 40
9.94
0.46
0.5T
1.18
6.28
UDE- MRVCJ
0.34
0.3G
0.54
T.85
0.40
O. 40
47
10.42
0.47
0.38
1.09
6.00
GARCH
1.91
1.21
1.60
14
2.89
GO
26
34.40
0.80
0.68
0.77
16. 76
EWMA
1.38
1.03
1.28
22.40
2.20
1.42
.292
20.77
0.48
0.54
0.64
14.74
FIGARCH
1.97
1.21
1.62
27.61
2.03
1. ..00
2 3.27
36.76
0.85
0.69
0.79
17.02
Merton31>
1:31
1.03
1:27
2.13
.11
28.45
0.42
0.53
D-50
12:31
LSTM
0.37
0.48
1.37
9-31
0.20
0.50
0.28
5.87
0.79
0.83
3.81
16.67
GRU
0.37
0.47
1.07
8-23
0.91
0:39
0:37
7-48
0.65
0.74
2.81
Time:PM
0.32
0.98
6.74
0.55
0.06
2.39
38.38
TimesPM
0.31
8.65
0.52
0.63
R100el12000
RW
0.58
Vanilla Nearal
0.51
0.61
11.36
1.49
17.82
UDE-ME
0.39
0.39
0.68
5.00
10.91
0.58
10.68
UDE MRYC
0.36
0.37
0.61
8.54
0.40
0.55
UDE- UDE-MEVOS MIVCJ
0.37
0.58
0.63
8.68
0.43
0.45
10.05
0.8.2
0.42
1.45
8.51
GARCH
1.27
0.98
1.22
22.98
2.15
1.41
L.DO
81.47
0.48
0.56
0.68
14.18
A
1.03
0.87
1.05
20-D1
1.97
1. 29
28.58
0.32
0.43
0.50
11:20
PIGARCH
1.18
0.94
1.13
21.82
2.06
1. 3.3
30.40
0.3T
0.18
D-47
12:42
MartonJDD
0.98
0.87
1.02
19.86
1.81
1 30
28, 40
0.41
0.32
0.42
10.81
LSTM
0.46
0.52
2.26
10.55
0.51
0. 44
6.70
0.10
0.35
6.05
8.16
GRU
0.47
0.5G
1.78
11.55
0.53
0. 09
72
12:21
0.70
0.68
4.10
14, 14,90
TMmePM
0.39
0.49
1.44
10.28
0.42
0.59
10.88
0.64
0.68
14.58
Time.PM
0.37
0.48
1.38
0.42
0.65
10.98
0.88
0.02
3.83
13.78
RW
0.34
0.39
7.68
0.97
0.82
0.46
1.88
ODE-MR
0.32
0.4G
9.05
0.68
0.74
0.82
0.4G
0.24
0.39
02
7.68
0.33
0.45
1.27
UDE-
UDE- MRVC
UDDE-MRVCS
0.24
0.24
0-42
7.21
0.25
0.25
6.54
0.32
0.32
0-42
7.61
GARCH
0.98
0.55
1.01
1.79
26
25.53
0.29
0.44
0-44
10.07
EWMA
0.97
0.83
0.97
17:31
1.59
24.95
0.32
0.59
0-43
FIGARCH
0.92
0.82
0.94
17.00
1.69
1.22
1.66
24.06
0.36
0.41
0.40
9.52
MertonJD
1.05
0.88
1.04
15.35
1.02
1.31
1.73
26.61
0.33
0.42
0.39
9.58
L8TM
0.32
0.56
2.26
11.02
0.46
0.64
0.88
10.18
0.88
0.78
5.80
15.91
GRU
0.52
0.57
1.47
11.40
0.64
0.64
0.72
12.45
0.64
0.65
3.49
13.50
Fime:PM
0.30
0.48
0.56
8.41
0.39
0.62
0.60
9.78
0.37
0.46
1.87
9.51
TimeePM
0.27
0.41
0.73
8.04
0:37
0.51
0:48
9-52
0.32
0:44
1:51
8.21
22

---

Published in Transactions on Machine Learning Research (12/2025)
Table Medium Horizon 5) Performance: Crypto & Bonds
Aggragated
High
MAE
QL
MAR
QL
QL
Bitcoin
BW
0.51
0.57
2.11
14.80
0.59
0.62
0.71
18.05
0.72
0.89
6.06
ODE-MR
0.50
0.84
3.00
19.04
0.51
0.61
0.42
14.30
1.45
1.16
9.15
30.78
Vanilla Neural ODR
0.42
0.51
1.29
13.20
0.54
0.60
0.08
14:45
D. 80
0.64
3.30
17.68
UDE-MR
0.50
0.53
1.22
12:31
0.34
0.61
0.48
8-40
0.72
0.43
20.63
UDE-MRVC
0.49
0.52
1.21
12:07
0.34
0.60
0.48
7-548
0:71
0.40
20-48
UDE-MRVCJ
0.47
0.50
1.14
11.78
0.57
0.59
0.44
7.08
0.68
0.
20.38
GARCH
1.11
1.07
25.07
2:12
1:38
1.85
88
0.37
0. 36
11.00
0.34
FIGARCH
1.01
MertonJD
0.24
9.80
L.TM
TimesFM
TimesPM PS
Rshereum
RW
ODR-MR
0.90
15.
37.45
Vanilla Neural ODE
0.32
12.56
0.42
8H5
2.13
UDE-MR
0.40
8 40
0.92
12.10
0.58
0.56
0.81
21 GO
4G
0.45
0.39
11.41
UDE-MRVC
0.30
0.39
0.90
11.58
0.56
0.06
0.31
21.23 21
4G
0.43
0.33
10.25
UDE-MRVCJ
0.38
0.38
0.88
11.75
0.54
0.54
0.32
20.04
0.4L
0.39
0.32
9.84
GARCH
0.83
0.77
0.88
23.26
0.19
0.34
0.35
11.74
L.BL
1.19
1.51
34.34
EWMA
0.08
0.87
0.76
20.18
0.18
0.63
0.38
11.22
1.84
1.07
1.31
30.44
RIGARCH
0.79
0.34
0.38
11: 733
1.15
Marten.I
0.70
0.89
03
1.43
1.39
31
130
0.28
TimesFM FE
ACG
8.97
ODE-MR
Vanila Neural ODE
0.29
0.42
6.53
0.48
0.32
7:351
0.42
1.41
8.34
UDE-MR
0.29
0.29
0.47
6.50
0.30
0.30
0.23
6.45
0.44
0.44
0.58
7.85
UDE-MRVC
0.28
0.28
0.43
6.85
0.31
0.51
0.30
7.50
0.44
0.35
0.48
6.20
UDE-MRVCJ
0.29
0.29
0.54
7-40
0.32
0.32
0.34
8.20
D-46
0-44
0.68
7.95
GARCH
1.61
1.17
1-43
20.38
2.62
1.56
1.98
26.322
0.72
0.77
13-42
EWMA
1.28
0.98
1.18
16.50
1.98
1.34
1.57
22.10 16
0.49
0.68
0.02
10.56
FIGARCH
1.51
1.11
1.38
19.30
2.39
1.49
1.84
24.05
0.G1
0.67
0.70
12.49
MerSonJD
1.36
1.05
1.33
18.00
2.25
1.45
1.76
24
0.52
0.62
0.62
11:19
TimeePM PS
23

---

Published in Transactions on Machine Learning Research (12/2025)
Table 9: Medium Horizon (h 5) Performance: Equities
Asgregated QL
Vol
High
Model
MSE
xMd
MAR
MEE
Equcties Dow.onne
RW
0.30
0-44
0.84
8.59
0.52
0.40
0.42
8.08
0.42
0.53
1.97
10.90
ODE-MR
0.54
0:62
0.04
12:45
1.30
0.99
1.10
10 .3.33
0.38
0.53
0.22
10.53
Vanilla Naural ODR
0.27
0-12
0:73
0.313
0:48
0.41
8-869
0-40
0.52
1.78
UDE-MI
0.29
0.29
0:58
0-313
0.31
0.62
5.06
0-41
0:41
0.51
9-48
UDE-MRVC
0.29
0.03
0.82
0.57
0.37
7.86
UDE-MRVCJ
0.33
0.06
5.87
1:50
EWMA
FIGARCH
10.04
1.43
9.07
MartonJD
0
0.80
16.38
1.17
1.48
23
0.24
0.12
0.38
8.84
LSTM
0.50
D.56
10.82
0.42
0.52
0.59
9.73
0.72
0.73
3.95
14.85
GRU
0.4B
0.54
1.27
10.74
0.53
0.5T
0.61
11 11.04
0.63
0.89
3.09
14.05
TimesFM
0.35
0.47
0.95
9.26
0.4G
0.56
0.37
10.70
0.45
0.00
2.14
11:37
TimesPM PS
0.34
D-46
0. 8888
9.11
0-46
0.57
0.36
10.75
0.41
0.52
1.93
10.82
PTSR:
RW
0.27
D-41
0.71
5.10
0.31
0.44
0.40
8.45
0.39
0.52
1-69
10.65
ODR-MI
0 1.49
D-60
0.59
12:01
0.97
0.91
1.05
18. 18.30
0.36
0.53
0.18
13:19
Youillaa Neural ODE
0.26
0.40
B.GG
7:97
0.33
0.46
0.42
8.88
0.37
0.49
L.5G
10.09
UDE-MR
0.27
0.27
0.44
7.06
0.30
0.50
0.51
10 62
0. S7
0.26
0.
6.86
IDE-MRVC
0.26
0.43
6.04
0.31
0.31
0.41
0. 37
0.28
0. 8.40
5.35
UDE-MRVCJ
0:27
0.27
0.91
0.41
O. 39
0.27
0.44
6.53
0.03
13:78
3:12
0.97
1.14
19.72
0.21
0.36
8.17
RWMA
PIGARCH
MartonJID
LSTM
1.25
0.35
0. 75
0.78
GRU
0.35
0.32
0.43
0.42
13.61
TimeaFM
0.26
0.40
0.71
7.92
0.30
0.45
0.40
8:46
0.41
0.54
L.T5
11.10
TimcsFM FS
0.26
0.40
0.69
7.79
0.30
0.45
0.40
8.65
0.38
0.61
1.01
10.58
NASDAQ
RW
0.30
0.44
0.78
9.05
0.85
0.48
0.46
9.49
0.38
0.01
1.01
11:21
ODE-MR
0.35
0.48
0.48
10.02
0.73
0.79
0.52
1G D8
0.36
0.01
0.40
11:18
Vanilla Neurol ODE
0.28
0.42
0.70
8.57
0.35
0.49
0.44
0.322
0.34
0:40
1.41
10.69
UDE-MR
0.29
0.29
0.59
8.98
0.34
0.34
0.61
12 12.85
0.36
0.35
0.49
8.32
UDE-MRVC
0.29
0.29
0.54
B.49
0.35
0.35
0.ST
6.B7
0.36
0.36
0.49
9.94
UDV-MRVCJ
1:30
0.30
0:53
8.77
0.35
0:35
0.57
11 TO
0.311
7.22
GARCH
0.81
1.10
20.96
2.02
1.38
1.80
28
O. 3.36
0-18
0-08
11.75
EWMA
8888
0.79
0.81
17-54
1:21
1.5/3
25
0.28
0.40
9-70
PIGARCH
1.05
L.05
1:74
0.
11.27
MartonJD
3:74
0.39
LSTM
GRU
14
TimesPM
0.64
11
0.64
11.86
TimedPM PS
0.36
1.05
0.49
0.60
0.63
11 96
0.
2.32
10.75
NIFTYD0
RW
0.44
0.39
8.40
0.49
1.63
9.99
ODE-MR
0.45
0.86
0.54
11.52
0.91
0.91
1.11
17.
0.48
0.15
9.86
Vamilla Noural ODE
0.30
0.39
0.58
7.87
0.30
0.44
0.38
8.49
0.35
0.47
1.81
9.48
UDE-MR
0 27
0.37
0.41
7.25
0.30
0.30
0.47
6.64
O. 0.36
0.25
0.43
5-13
UDE-MIVC
0 26
D.26
0.39
13
0.31
0.31
0-43
8.55
0.35
0.25
0-12
5.05
UDE-MIVCJ
0.25
0.25
0.358
7.08
0.51
0.31
0-42
0.34
0.24
0.33
5.01
GARCH
1:05
0.89
1.04
19.19
1.81
1.26
1.63
2G 16
O. 34
0.47
0.44
11,00
EWMA
83
0.76
0.85
16.23
1.43
1.31
1:36
22 TO
0.28
0.40
0.
9.38
PIGARCH
17
0.86
18.
3/70
1.29
1:56
25 34
O. 30
0:45
0.
10:40
Merton.ID
0.91
17.08
1,80
1.15
1.44
28
0.28
0.43
9.81
LSTM
0.40
1:25
0.28
0.39
0.
0.79
3.
15.97
TimeePM
TimesPM
Raasell100
RW
ODE-MR
1.05
Vanilla Noural ODR
0.26
0.39
0.80
7.51
0.25
0.40
0.37
7.47
0.54
1,92
11.02
UDR-MB
0.28
0.30
0.48
10
0.29
0.51
5.06
UDE-MRVC
0.27
0.27
0.02
7.81
0.20
0.29
0.45
9.81
0.40
0.28
0.49
5.02
UDE-MRVCJ
0.27
0.27
0.50
7.81
0.30
0.30
0.44
0.61
0.41
0.28
0.47
6.01
GARCH
1:41
1.06
1.32
23.02
2:30
1.46
1.98
30 64
0.47
0.54
0.53
12:70
EWMA
1.25
0.95
1.18
21.11
2.06
1.37
1.83
28. T3
0.39
0.46
0.46
11.35
FIGARCH
1.39
1.05
L.30
22.82
2.22
1.43
1.94
30 .0
0.46
0.63
0.52
12.51
MartonJID
1.29
1.25
92.06
2:15
1.42
1.92
28 TO
0.36
0.51
0-97
11 .666
LSTM
0.38
0.19
0.22
0.32
0.35
5.98
0.85
4.18
17.25
GRU
0.58
L3
0.33
0.41
0-42
7:76
69
0.76
3. 0.08
15.56
TimesPM
0.32
0.42
0.00
0.33
0.42
11.92
Raceel12000
0.46
Vonilla Neural ODE
0.44
9.24
UDE-MR
0.36
0.51
0.39
UDE-MRVE
0.30
0.36
0.51
11 16.86
0.37
6.54
UDE-MRVCJ
0.30
0.30
0.50
8.33
0.37
0.ST
0.50
11. 14.14
0.ST
0.32
0.47
7.44
GARCH
1.07
6.90
1.08
20.70
1.87
1.51
1.71
28 88
0.38
0.46
0.48
11.69
KWMA
0 8.80
D.81
0.84
18.50
1.63
1/21
1.54
26 .12
0.25
0.39
0.39
9.65
PIGARCH
0.95
D.85
G. 99
19.44
3.71
1.25
1.61
27 A06
0. 0.27
0.11
0.90
10.25
Marton.ID
0.95
D.NS
G. 89
19.38
3.75
1/27
1.69
27 WT
0.50
0.46
0.36
9--45
LSTM
0.44
0.62
2. 31
10.07
0.81
0.43
0.54
0. 58
0.92
5.46
17.62
GRU
0.63
17:27
0.52
0.58
0.70
12 11
0. G4
0.65
3.
14. 18
TimonFM
0.33
L.04
0.41
0.54
0.65
10.0
0.56
12.44
TimcsFM
0.95
0.40
0.54
0.54
0.38
2.22
11.74
8P600
0.98
0.88
0.50
0.40
0.68
2.08
11.29
ODE-MR
18.07
1:24
1.05
1.20
0.48
0.20
10.24
0.44
0.37
10
0.54
UDE-MR
UDE-MRVC
UDR-MRVCJ
0.34
0.34
0.61
9.16
0.35
0.65
12
0.
0.
6.39
GARCH
D.84
1.14
19.93
2.13
1.38
1.88
28
0.34
0-17
0.4K
10.85
EWMA
D.85
17.78
1.28
1.68
25
0.28
0.40
0.64
9.22
FIGARCH
1.15
0.92
1.11
19.51
2.08
1.37
1.82
27.77
0.34
0.46
0.45
10.74
MoctonJD
1.06
0.89
1.05
18.56
1.00
1.33
1.76
26.81
0.20
0.44
0.40
0.68
LSTM
0.52
0.86
2.25
11.01
0.48
0.34
0.6T
10:20
0.88
0.80
5.72
16.28
GRU
0.52
0.67
1.46
11.41
0.65
0.65
0.72
12.45
0.84
0.86
3.50
15.58
TimecFM
0.41
0.61
1.10
9.98
0.50
0.63
0.67
11 16
0.46
0.66
2.40
11.30
TimesPM P8
0.40
0.50
1.04
0.55
0.61
0.67
12.02
0-43
0.53
2.21
10.94
24

---

Published in Transactions on Machine Learning Research (12/2025)
Table 10: Medium Horizon 10) Performance: Crypto & Bonds
Aggragated
High
MAE
QL
MAR
QL
QL
Bitcoin
BW
0.57
0.80
2.28
15.38
0.69
0.68
0.78
16.72
1.24
0.54
14.33
ODE-MR
0.00
0.84
3.04
16.10
0.50
0.62
0.44
14.46
0.92
0.87
9.12
18.43
Vanilla Neural ODR
0.40
0.53
1.68
13:50
0.60
0.64
0.70
15-53
0. 7.
0:73
4.90
18.05
UDE-MR
0.35
0.48
1.25
12:38
0.05
0.52
0.30
9-10
7.75
0:73
0.40
10:39
UDE-MRVC
0.35
0.48
1.23
12:28
0.67
0.47
0.47
8.12
0-74
0:72
0.40
10.15
UDE-MRVCJ
0.52
0-47
1.16
11.01
0.66
0.46
0.45
8.12
0.73
0.71
0.39
9.98
GARCH
1:26
0.95
1.88
27.22
2:38
2.00
49
0.77
0.74
20:57
FIGARCH
0.37
MertonJD
L.TM
TimesFM
TimesPM PS
Rshereum
RW
0.46
0.25
ODR-MR
23
Vanilla Neural ODE
0.31
0.47
0.55
0.55
UDE-MR
0.40
8.44
0.96
12.19
0.05
0.46
0.40
11:04
0.G8
0.67
0.31
21.74
UDE-MRVC
0.44
8.44
0.91
12:0T
0.53
0.41
0.34
10.08
02
0.65
0.31
21.29
UDE-MRVCJ
0.42
0.42
0.90
11.0T
0.52
0.44
0.34
10-47
0.50
0.55
0.51
20.21
GARCH
0.94
0.82
0.96
25.08
1.80
1.28
1.64
36.72 722
8.59
0.59
0.55
12.40
EWMA
0.78
0.89
0.78
20.77
1.43
1.10
1.30
91
02
0.65
0.36
10.57
RIGARCH
0.84
0.80
40
Marten.I
0.71
0.77
1.40
TimesFM FS
ACG
ODE-MR
7:85
Vanila Neural ODE
0.33
01.6
0.80
7.15
7.70
0.44
0.84
1.48
8.75
UDE-MR
0.80
0.30
0.52
6.35
0.35
0.35
0.26
6.95
0.45
0.45
0.85
7.95
UDE-MRVC
0.31
0.31
0.95
6.90
0.37
0.37
0.53
7.70
0.45
0.39
0.52
6.55
UDE-MRVCJ
0.33
0.32
0.63
7.55
0.39
0.38
0.12
8.70
46
0-46
0.83
8.50
GARCH
1.95
1.28
1.83
22.35
3.00
1.67
2.18
28-44
D.BL
0.84
0.97
15.88
EWMA
1.30
1.01
1.10
17.32
2.09
1.37
1.64
22.81
0.6L
0.61
0.70
10.99
FIGARCH
1.70
1.21
1.51
21.26
2:76
1.59
2:04
26.03
0.79
0.77
0.86
14.53
MerSonJD
1.38
1.07
1.25
18.21
2.28
1.46
1.78
24 16
0.54
0.64
11.52
6.74
6.98
TimeePM PS
25

---

Published in Transactions on Machine Learning Research (12/2025)
Table 11: Medium Horizon (h 10) Performance: Equities
Aggeegated
Model
MSE
MAE
MSE
xM
Bysdiies
DowJonne
RW
0.56
0.83
0.64
12.55
1.12
1.01
1.18
19.80
0.47
0.56
0.43
11.02
ODE-MR
0.54
0.46
1.00
9.03
0.38
0.49
0.67
9-35
0.52
0.58
2.86
11.70
Vauilh Neural ODE
0.38
0.45
0.98
8:93
0.39
0.51
0.50
9.63
0.46
0.68
2.32
10.85
UDE-MR
0.33
0.33
9-01
0.37
0.37
0.64
5.48
D-31
0.53
5.45
UDE-MRVC
0.32
0.32
8.87
0.37
0-37
0.59
11.84
0:36
8-35
0:51
8.07
UDE-MRVCJ
0.34
8.07
11.21
0.82
0.55
1.65
0.64
0.53
FICARCH
13.74
MerSond
0.58
0.79
0.91
1.61
13:02
LSTM
0.53
0.59
2
13:35
0.45
0.54
10.05
0.76
4.26
15.62
GRU
0.45
0.34
10.79
0.64
0.58
0.61
11.17
0.64
D.TO
3.09
14.28
TMme=FM
0.38
0.49
1.08
9.68
0.47
0.58
0.57
10.76
0.52
D.BO
2.58
12.34
TimesPM PS
0.36
0-48
O 99
9.48
0:47
0.37
0.57
10:84
0.48
D.
3.32
11.60
FTSE
RW
0.30
0.43
0.82
8.58
0.85
0.48
0.45
9.06
0.47
0.68
2.18
11.89
ODR-MI
0.50
0.61
O. 6O
12:16
1.00
0.95
1.08
18.70
0.42
D.52
0.18
10.82
Vandla Neural. ODE
0.35
0-47
O. 96
9-41
D-42
0.54
0.55
10:35
0.46
D-54
2.12
11:38
UDE-MR
0.20
0.29
0.47
7-57
0.34
0.34
0.53
5:06
0.39
D. 0.85
0.44
8.63
MRV-
0.28
0-28
0.44
7:23
0:34
0-34
0.43
8.50
0.35
0.34
0.43
5.60
UDE-MRVCI
0:30
0:30
0.48
7-16
0:35
0.35
0.42
B.
0.42
0.31
0.43
5.40
GARCH
0.88
14.45
1.23
11.84
0.62
12.98
1.07
79
12:0T
Marton.D
GRU
0.34
0.47
13.74
Timed/M
0.30
0.42
5-17
0:30
2.02
11.51
TimesFM FS
0.28
0.41
0.63
8.05
0.30
0.45
0.40
8.47
0.47
0.66
1,80
10.85
NASDAQ
IOW
0.31
0-47
0.99
9:71
0.40
13.34
D.61
12,62
ODE-MR
0.36
0.49
0.99
10.18
0.75
0.79
0.84
16.85
0.50
0.00
0.48
12.79
Vanilla Neural ODE
0.36
0.47
1.28
9.74
0.45
0.55
0.00
10.01
0.48
0.56
2.12
11.90
UDES-MR
0.33
0:33
0 65
9.28
0.42
0.44
12:01
0.43
0:41
0.51
5:92
UDE-MRVC
0.32
0.52
0.63
9.07
0.59
0.39
0.59
7.16
0.39
0.37
0.50
T.49
UDE-MRVCJ
0.38
0.35
0.02
9.03
0.40
0.40
0.59
12.05
0.97
0.98
0.52
7.78
1.27
0.88
1:21
22.45
2:32
31 89
0.69
0.51
13.78
0.92
0.81
18-11
80
13.32
FIGARCH
1.29
0.95
2:27
31 25
0.51
13.90
MertondD
GRU
14.13
TimesFM
12.12
TimesFM
0.50
28
12.51
N0PT
RW
8.33
10.62
ODE-MR
0.45
0.57
53
63
11.12
Vanilla Neural ODE
0.29
0.41
0.59
8.01
0.94
0.48
0.46
9.20
0.39
1.79
9.89
UDE-MR
0.30
0.30
0.44
7.50
0.85
0.33
0.48
10.62
0.39
0.35
0.45
7.31
UDREMRVV
0.28
0.28
0-42
7:23
0-32
0.32
0.44
9.09
0.37
D:31
0.45
6.01
UDE-MRVCJ
0.27
0.27
0.40
7:22
0.55
0.31
0.42
W.97
0.34
0.35
0.35
5.12
GARCH
1:21
0.96
1.16
20.55
D:D0
1:35
1.766
27 DIS
0.64
D.70
0.57
13:42
EWMA
0.85
0.78
0 88
16.00 06
4.
3.14
1.42
23 44
0.09
0.74
0.38
12:45
PIGARCH
1.10
0.81
00
19.54
28
26 TO
0.72
0.76
0.52
13.74
Merton31>
0 86
0.80
17-18
1.51
80
0.71
73
0.39
12.88
0.49
9.82
0.53
8.74
16.29
CRU
0.61
0.45
0.52
9-81
2.87
TimeFM
TimeFPM
Ramelll1n0
BW
ODE-MR
12.98
Vanilia Neurol
0.34
12.48
UDERMR
0.31
0.31
0.61
8.10
0.33
0.50
10.58
0.35
0.52
7.01
UDE-MRVC
0.30
0.30
0.57
8.06
0.32
0.47
10.04
0.40
0.31
0.52
6.01
UDE-MRVCJ
0.31
0.31
0.65
8.02
0.31
0.31
0.45
8.75
0.30
0.90
0.02
6.87
GARCH
1:63
1:13
4.45
24.59
2.08
1.53
32
0.67
0.74
0.61
14.04
RWMA
1.27
0.89
1 11
31
2.10
1.39
188
26 09
0.64
D-70
0.46
12:33
FICARCH
1.06
1.13
1.48
24.07
2.68
1.53
2.12
32.07
0.70
D.76
0.03
14.25
MarionJD
1.30
1.02
26
22.05
2.15
1.42
1.02
25.68
0.98
0.74
0.46
12.82
0.38
0.49
9.48
D-40
0.45
0.50
8-12
0.84
0.22
17.07
GRU
0.38
0.48
1.14
9.41
0:43
7.74
0.67
D.75
3.07
15.12
TimeFPM
0.44
8.74
0:95
8.30
0.54
2:43
13.10
3.23
Rasel12000
RW
0.47
9.98
2:85
12.67
Neural
UDE-ME
0.35
5.70
UDE-MRVC
0.38
0.33
0 56
UDE-MRVCJ
0.32
0.32
0.54
8.52
0.37
0.88
0.46
6.74
GARCH
1.16
0.94
1.15
21.84
2.06
1.37
1.85
30.64
0.87
D.T5
0.50
13.28
EWMA
0.96
0.84
0.98
19.07
1.75
1.25
1.01
27.64
0.70
0.75
0.30
12.63
PIGARCH
1.06
0.80
1-07
20.61
L.9
1.31
1.72
20.03
0.74
D.77
0.33
13:08
Merton.D
0.97
0.86
1.00
19:64
L.79
1/29
1.67
28 21
0.72
D-74
0.38
12.85
LSTM
0.44
0.52
2. 2.10
10.67
D.42
0.47
B-T4
0.87
D.92
5.34
17.59
GRU
0.
0.54
GO
13.34
0.63
0.59
0.73
12.21
0.64
0.64
3.68
14.08
TimesPM
0.47
19
9-76
0:42
0.54
0.58
10 82
1.53
2.89
13.13
TimesFM
0.46
9.58
0.42
10.98
0.50
0.67
2.65
12.05
RW
0.65
14.02
1.26
1.27
20.62
3.98
12.80
ODE-MR
68
0.66
0.
18.23
16.03
11.58
Neural
UDE-MR
UDE-MRVC
0.38
0.38
0 73
0.41
12.86 86
0.37
6.07
UDV-MRVCJ
0.39
0.39
0.69
9.49
0.44
0.44
19.54
D.43
GARCH
1.30
1.00
30.28
11.01
EWMA
1.06
0.87
1.05
18.25
1.08
1.31
1.72
26.65
0.20
0.41
0.46
9.38
FIGARCH
1.31
0.98
1.21
20.98
2,30
1.44
1.06
29.02
0.37
0.47
0.47
11:09
Marton.D
1.08
0.89
1.07
18.75
1.58
1.34
1.77
27.10
0.35
0-45
0.41
9.78
LSTM
0.32
0.56
2.18
11.02
0.46
0.55
0.66
10.52
0.87
0.79
5.57
16.07
GRU
0.52
0.57
1.45
11.45
0.64
0.65
0.71
12.44
0.86
0.07
3.40
13.84
TimesPM
0.44
0.53
.6
10.36
0:57
0.63
0.68
11.66
0.54
0.59
2.82
12:10
TimedPM
0.43
0.52
10.25
0.57
0.64
0.68
13.03
0.51
0.58
2.73
11.97
26

---

Published in Transactions on Machine Learning Research (12/2025)
Table 12: Medium Horizon 20) Performance: Crypto & Bonds
Aggneggate QL
High
Auset
QL
MBB
Bitcoin
RW
0.57
0.80
2.50
15.58
0.65
0.55
0.27
16.08
0.96
0.76
6.53
20.65
ODE-MR
0.60
0.04
3.06
16.10
0.88
0.68
0.42
15.36
1.44
1.15
9.10
30.43
Vanilla Neural ODR
0.50
0.56
2. 2.20
14.337
0.70
0.68
0.93
16.65
0.73
0.69
6.06
18:65
UDEE-RR
0.55
D.47
1.19
8.43
0.65
0-45
0.45
9:27
0.80
0.73
3.47
20.38
UDE-MIVV
0.54
0-48
1.27
12.60
0.64
0.52
0.50
9-30
0.
0.74
3.67
20.54
UID-MRVCC
0.50
0.49
1.26
12:50
0.61
0.51
0.48
0:85
0.76
0.
3.56
19.93
GARCH
1.48
1 03
31
30.00
1:55
2:17
42 33
0.36
46
0.45
15.46
0.57
FIGARCH
1.26
0.38
0.45
MertonJD
1.28
38.
GRU
TimesPM
5a
13.56
3.60
TimesPM PS
18.04
Rchereum
RW
0.48
53
63
19.25
ODR-MR
83
21.43
0.43
36
15.27
Vantlla Neural ODE
0.61
15
2.18
17.65
UDE-MR
0.46
0
0.94
12.70
0.59
0.42
0.36
10.24
0.60
0.00
2.07
21.39
UDE-MRVC
0.45
0.64
90
12.30
0.58
0.47
0.42
11
0.57
0.07
2.86
21.72
UDE-MRVCJ
0.43
0.43
0.91
12:45
0.86
0.42
0.35
10:30
0.56
0.55
2.51
19.93
GARCH
1.12
0.
1.00
27.90
2.07
1.35
1.80
39.85
0.30
0.45
0.44
15.15
EWMA
0.50
0.78
0.54
22.08
1.80
1.17
1.48
33.30
0.18
0.58
0.34
11.34
PIGARCH
1:06
27.39
1:31
3W 63S
0-41
0-43
14.74
MartonJID
1.32
1:18
32 52.55
0.32
10.32
2.13
1.28
Time-M
Bonds
XCG
ODE-MR
7.45
7.95
Vantlla Neural ODE
0.35
0.47
7.30
0.40
0.51
0.37
7.80
0.48
0.86
1.52
8.95
UDE-MR
0.28
0.28
0.54
8.50
0.33
0.33
0.28
0.41
0.41
0.72
8.05
UDE-MRVC
0.29
0.29
0.58
7.05
0.35
0.35
0.35
7.75
0.23
0.36
0.57
6.52
UIDE-MRVCJ
0.30
0.
0.67
8.10
0.37
0.37
0.46
8.95
0.22
OLAVT
D.33
6.40
GARCH
2:41
1-42
1.99
25.40
3.36
1.81
2.45
31.15
1.24
0.97
1-22
18.70
EWMA
1.42
1. .606
1.28
18.30
2.25
1.44
1.75
28.95
0.64
0.67
0.88
12:23
FIGARCH
2.17
1 .3
1.78
23.67
1.70
2.24
20 29.06
1.08
0.8G
1.05
16.41
MertonJD
1.43
1 88
1.28
18:47
2:33
1.47
1.80
24.40 80
0.57
0. 05
0.71
11.76
9.29
0.84
0.41
0.78
GRU
7.42
0.58
TimaPPM
27

---

Published in Transactions on Machine Learning Research (12/2025)
Table 13: Medium Horizon (h 20) Performance: Equities
Aggregated
Low
Vol
Model
MSB
MBE
Equitiis
Dow Joncx
RW
0.38
0.50
1.20
9:75
1.09
0.99
1.1.
19. 02.32
0.59
0.84
3.20
13. 13.01
ODE-MR
0.55
0.63
0.65
12.56
0.43
0.52
0.52
9.81
0.52
0.82
0.27
13.00 13.06
Vanilla Nenral OTE
0.48
0.55
1.48
10:90
0.33
0.60
11-43
0.63
0.95
3.68
13.
0.37
0.37
0.88
8:22
0.38
0.33
B.10
0-53
0:45
1.32
0.04
0.38
0.38
0.52
1.40
7:30
MREC
8:85
1.0
1:16
1-13
0.00
EWMA
PICARCH
1.02
1.01
Merton.]I
0.88
0.79
0.92
16.70
59
23
0.39
0.51
0.42
9.31
LSTM
0.55
0.63
2.48
11.87
0. 9B
0.56
0.
10.
0.80
5-02
16. 16.78
GRU
0.48
0.54
1.58
10.86
0.51
0.56
0.
10.00
0.65
0.58
3.334
14.14
TimesFM
0.40
0.50
1:27
9.88
0:45
0.64
0. 0.56
10.40 40
0.38
0.64
3.12
13.18
TM PS
0.38
0.50
1.18
9.82
0:45
0.55
0. 56
10.49 10.47
0.55
0.61
2.88
12.50
PTSR
RW
0.33
0.46
0.86
8.99
0-41
D.51
0. 50
8-72
0.48
0.58
2.20
12.14
ODE-MI
0.51
0.81
0.81
12.30
1-01
D.96
1. 1.10
85.85
0-16
0.53
0.16
13-13
Vauilla Neural ODE
0.00
0.61
2.07
12.31
0.67
0.67
0.94
13. 30
0.78
0.69
5.44
14.10
UDE- MR
0.32
0.32
0.49
5.12
0.
0.39
0. DG
11. 30
0.45
0.32
0.94
5.30
UDE- MRVC
0.30
0.30
0-44
7:30
0 4-
D- 40
0 45
7-78
0.44
0.31
1.04
5.23
CRV
0.33
0.38
0.44
7-22
0-42
0.47
0.44
0.99
9.18
0.79
0.78
0.83
1.08
22
0.24
0.40
8.87
PWWA
PIGARCH
Marton
LSTM
0.34
1.24
0.28
0.37
0.36
0.78
3.39
15 64
GRU
0.34
1.00
8-22
0.60
0.59
2.70
13
FimeFM
0.29
0.43
0.85
8.39
0.30
D.45
0.41
8.44
0.51
0.62
2.26
12
FimcsFi FS
0.28
0.42
0.81
8.28
0.31
0.45
0.41
8.54
0.49
0.59
2.14
12, 22.21
NASDAQ
RW
0.38
0.49
1.26
10.04
0.41
D.
0.62
10.47
0.68
0.07
8.43
14 38
ODE- MR
0.37
0.50
0.51
10.34
0.78
0.79
0.93
10.76
0.56
0.60
0.50
14.
Vanilla Neural ODE
0.64
0.58
3.50
11.80
0.69
0.68
1.00
13.
0.93
0.06
9.90
13.90
UDE-MR
0.59
0.39
0.72
9.80
0-42
D-42
0.
13. 06
0.58
0.64
1.86
8:51
UDE-MRYE
0.38
0.30
0.69
9.26
0.39
0.59
0. 0.59
12.
0.55
0.41
1.63
8.01
CVV-
0.37
0.37
0.71
8:39
0:81
D-41
0. 59
0.60
0.42
1.57
8:71
GARCH
1.47
1.07
1.33
21.72
2.52
1.53
0.50
0.58
0.59
0.98
0.84
0.98
18.83
1.27
1 85
27
0.31
0.43
0-45
10
FIGARCH
1.46
1.30
0.56
0.30
MD
0.95
0.43
0.56
L
GRU
0.35
1.08
4.74
TimcsPM TimerPM PS
0.45
1.64
12.
4.05
0.43
0.52
1.50
0.51
0.83
0.
12.
0.60
0.52
3.70
15
NIPTY50
RW ODE-MR
0.31
0.5G
8.73
0.47
0.56
11.51
0.45
0.57
0.55
11.41
0.92
0.91
1.02
17.00 97.
0.45
0.80
0.39
12.44
Vanilla Neural ODE
0.40
0.48
1.67
9.36
0.48
0.56
0.86
10.36
0.58
0.50
4.12
11.24
UTSE-ME MR
0.31
0.31
D.43
7-51
0.34
0.34
0.
6.42
0.46
0.37
0.82
7.14
UTDE UTVEMBVC
0.28
0.28
0-48
7-41
0.35
D.35
O.
9.84
0.11
0.41
1.OS
9.11
UDE-MIVCJ
0.28
0.28
0-4T
7-35
0.
D-36
0-333
0-40
0.28
D-99
5.14
GARCH
1.41
1.08
1.30
28.03
2.18
1.35
1.96
20. 36
0.64
0.63
0.09
15.42
EWMA
0.92
0.81
0.93
17 17.46
1.17
1-47
24
0.30
0.42
0.39
9.74
PIGARCH
1:28
0.87
1.20
21.52
00
1.32
27 80
0.55
0.58
0.61
13 80
Merton
0.88
0.03
1.17
0.29
0.44
0.42
10. 10.17
LSTM
0.40
1.40
9.86
0.27
0.47
0.38
0.78
0.89
8.01
10
TimesFM
TimesFM
Ramell1000
ODE-MR
Vanilla Nenral ODE
0.51
0.54
3.31
10.01
D.51
0.
8.26
1.08
0.75
9.70
16.36
OTDE-MR
0.35
0.50
7.89
0.38
38
0. 50
0.47
0.33
1.00
UDE-MRVE
0.32
0.34
0.53
7.15
0.36
D.
D.
6.49
0.44
0.44
1.29
9.86
EIDE-MRVCJ
0.20
0.91
0.5G
0.01
0.39
0.39
0.45
9.77
0.4G
0.46
1.14
9.48
GARCH
1.91
1.21
1.60
27.14
2 2.99
1.60
2 26
34 49
0.50
0.09
0.77
76.75
EWMA
1:38
1:03
1.28
22
2.20
1.42
L.22
29. 77
0.45
0.54
0.54
12-74
FIGARCH
1.97
1.21
1.62
27.61
2
L.60
2. 27
29. 76.76
0.55
0.89
0.79
17 02
Merton.I)
1.31
1.00
1:27
1-41
0.42
0.53
0.50
12.31
LSTM
0.37
0.48
1.57
8:31
0.20
0.30
0.78
0.83
3.81
18
0.37
0.47
1.07
8:23
0.32
0.
0.85
0.74
2.81
15. 11
TimesPM
0.32
8:74
0.06
2.30
2:24
Reses12000
Vanilla
ODE
11.
UDE-ME
0.38
0.61
0.62
1.40
10.76
UDE- MRVC
0.35
0.62
0.40
63
5.80
0.55
0.38
1.43
UDE-MEVCJ
0.35
0.35
0.63
8.32
0.43
D.45
0.53
0.51
0.48
3.42
10. 10.50
GARCH
1.27
0.98
1.22
22.98
2.18
1.41
L.DO
31.47
0.48
0.56
0.88
16.16
FW A
1.03
0.ST
1.05
20-01
1.87
1.29
1.69
28 58
0.33
0.45
0.30
11.20
PIGARCH
1:18
0.91
1.13
21.82
2. 2.06
1.36
1 L.82
30 43
0.3T
0.48
0.47
12.42
MertonJD
0.99
0.87
1.02
19.86
1.81
1.30
1. B.
28. 433
0-41
0.52
0.42
10.81
LSTM
0.46
0.52
2:35
10.55
0.81
0-44
0. 66
5.70
0.10
0.36
5.00
8.16
GRU
0.47
0.56
1.73
11 11.55
0.
59
0. 72
12:3 21
0.70
0.68
4.19
14 84
TimmFM
0.39
0.49
1.44
0.54
10.
0.64
0.88
8.61
14.
TimmsFi
0.37
0.48
1.33
0.64
10.
0.58
0.82
8.33
0.48
0.54
1.74
0.64
10.
0.75
0.72
4.77
ODE-MR
0.03
0.60
0.02
0.07
0.88
1.41
5.70
0.94
UDE- MR
UDE- MEVC
UTDE MRVCJ
0.45
0.44
0.78
9.76
0. 6.67
12.
0.68
0.48
GARCH
1.52
1.06
1.35
23
2.61
2.
0.46
0.54
0.34
EWMA
1.12
1.08
2/02
1.34
27.
0.30
0.42
0-44
9.73
FIGARCH
1.31
1.05
1.33
22.73
2.56
1.50
2.
81.55 31.30
0.49
0.54
0.54
18.98
MertonJI
1.09
0.30
1.08
18.86
1.96
1.33
1.76
27.00 27.08
0.47
0.56
0.43
18.18
LSTM
0.53
0.37
2.41
11.17
0.43
0.55
0.84
10.00
0.98
0.84
6.51
17.06
CRU
0.52
0.58
1.59
11.68
0.00
0.65
0.05
12.03
0.72
0.70
4.05
14.05
TimoFM
0.48
0.54
1.51
10.73
0.64
0.61
0.06
11.50
0.65
0.86
3.84
13.61
TimewIM
0.45
0.54
1.42
10.60
0.62
0. 0.86
11:70
0.61
0.53
3.37
13.16
28


Reviews:
[{"review_id": "Bs8bQoVySQ", "reviewer": "Reviewer_SxSR", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The authors applied Universal Differential Equations (UDE)\nto financial volatility forecasting,\nand compared its performance against multiple models,\nincluding Random Walk, GARCH, LSTM, ODE, and even Transformers.\nThe experiment results show the superiority of the UDE methods,\nwhile also being relatively light-weight and fast.\n\nPros:\n1. Novel application of UDE in financial volatility forecasting,\nespecially bridging domain knowledge by imposing the structural constraint\nwith the flexibility of NNs.\n2. The UDE models show superior performance in the volatility forecast,\neven compared to the Transformer models, while only having \na fraction of the parameters of the transformers \nand running comparably fast with all other methods.\n3. UDE models show long range (multi-step rolling ahead) forecasting stability,\nwhich is important if we looking beyond just one-period forecasting.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors provide clear evidence to the results of the experiments.", "audience_yn": "Yes", "audience_explanation": "This paper applied UDE in the financial volatility forecasting setting,\nwhich will be at least relevant to some ML people\nwho are also interested financial applications.", "requested_changes": "1. I manually checked all the references and the following cannot be found\n(6 out of 28): \n\n[4] Federico Bucci. Deep learning volatility: A finance application. arXiv preprint arXiv:2006.10463, 2020.\n\n[5] Laurent Callot and Ekimetrics Research Team. Sales forecasting with neural stochastic differential\nequations. Ekimetrics Research Blog, 2022.\n\n[7] Ting Chen, Zhi Chen, and Bin Han. Hybrid models of garch and neural networks in financial volatility\nforecasting. Expert Systems with Applications, 176:114908, 2021.\n\n[9] Miles Cranmer, Alvaro Sanchez-Gonzalez, Peter Battaglia, Rui Xu, Kyle Cranmer, David Spergel, and\nShirley Ho. Discovering symbolic models with differentiable physics. Nature Communications, 11(1):1–9,\n2020.\n\n[26] L. Xie, W. Zhang, and Y. Li. Energy demand forecasting with neural ordinary differential equations.\nIEEE Transactions on Smart Grid, 10(6):6789–6799, 2019.\n\n[27] Dacheng Xiu and Siem Jan Koopman. The stochastic volatility model: A review. Journal of Econometrics, 187(2):433–440, 2015.\n\n\nFor example, Dacheng Xiu is well known for his work on financial econometrics,\nbut he didn't publish on the Volume 187 Issue 2 of Journal of Econometrics.\nMoreover, the pages 433–440 spans across two on articles on that issue \n(JOE link referred below) [1].\nThe situation is similar for other ones I mentioned too.\n\nI think this is a serious issue in terms of academic integrity,\nbut I would like to hold on to my suspicion as to why this is happening\nuntil getting a proper response from the authors.\nAlthough I quite like the idea of this paper and all results look \nvery promising, once I cast doubt on the integrity of the paper,\nI feel less confident in what I see being reported in the article.\n\n\n2. The GARCH models are fit by directly choosing GARCH(1,1) process,\nwhich should work surprisingly well most of the time.\nBut I would like to see a few more models (in terms of the flavors of GARCH),\nsome model selection \n(\"hyper-parameter tuning\" for GARCH so to speak, even though\nthis is definitely not what is called in the financial econometrics literature,\nbut searching for the best order of GARCH is just like grid-search),\nand more importantly ARMA-GARCH models.\nARMA-GARCH model will model the mean process and volatility process\nin the same time, while GARCH by itself only models the volatility.\nARMA-GARCH is not available in the python \"arch\" package,\nso I would recommend R package \"rugarch\". \nIn fact, for all GARCH-related modelling, using \"rugarch\" is better choice.\nThose additional models will definitely strengthen the analyses\nby showing the UDE approach is superior than several decades of development on\nfinancial econometric models (Robert Engle won Nobel prize for ARCH).\n\n3. It is well known in the financial research community that Yahoo Finance\nis not a reliable source of data.\nYF is certainly open and accessible, but calling it \"high-quality\"\nas the authors did in the introduction might be misleading\n(one example of discussion referred below) [2].\nI do acknowledge that the issue with YF should not be a very huge concern \nfor this study *per se*, as the assets of interest \n(mostly well-known market indices, plus two important crypto coins)\nshould be error-free, but I still think it would be worth it to replicate \nthe results on some other data source as a robustness check.\nThe gold standard in financial research is WRDS from Wharton,\nbut any reputable financial data provider would work too.\n\n[1] https://www.sciencedirect.com/journal/journal-of-econometrics/vol/187/issue/2\n\n[2] https://quant.stackexchange.com/questions/942/any-known-bugs-with-yahoo-finance-adjusted-close-data", "broader_impact_concerns": "I don't have any concern regarding the impact of the idea of this paper,\nbut I do have concerns regarding the aforementioned reference issue,\nand its potential broader impact to the community.\nI would like to see the authors' response.", "additional_comments": "Overall, the idea of the paper is solid, and the results are promising,\nbut I would like to point out to the editor regarding the issue\nof \"fake references\" and how proceed.", "extra_scores": {}, "extra_text": {}}, {"review_id": "hG5HF8qETW", "reviewer": "Reviewer_a1qS", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper presents an extension of the Universal Differential Equations (UDEs) approach for the financial domain, focusing on the volatility forecasting task. Universal Differential Equations introduce additional, domain-specific constraints into the learning process. The authors model the log-volatility of financial data by using two small feedforward networks, one for the mean reversion speed and the other for the long-term level. The final computation of the two values is adjusted to satisfy a set of constraints (e.g., mean reversion coefficient must be positive, long-term level must be bounded, etc).\nThe evaluation is conducted on a set of data gathered from 10 financial indices. The UDE approach is compared against statistical models, (Neural) ODEs, and foundation models for time series. The results show that the proposed approach achieves the best results across several forecasting horizons. The ablation study allows to understand the impact of both neural networks in the learning process.\n\nWhile providing strong empirical results, the paper would benefit from a careful round of rewriting, as the presentation quality can be largely improved (see comments below).", "claims_and_evidence_yn": "Yes", "claims_explanation": "The empirical evaluation consistently shows that the UDE version proposed by the authors achieves excellent results in the forecasting of the volatility across different prediction horizons. The performance gain with respect to to other approaches like Neural ODE is largely due to the domain-specific knowledge introduced in the training process, as shown by the ablation study.", "audience_yn": "Yes", "audience_explanation": "The paper is of interest to researchers combining machine learning approaches for financial data. \nHowever, there is little chance that people not familiar with the financial domain will find useful insights from this contribution. This is not a negative aspect per se. It just reflects the domain-specific nature of the paper: while the high-level idea of injecting domain-specific knowledge in the training process is easy to understand, the specifics of how this has been done in the paper are only really accessible to researchers with a financial background.", "requested_changes": "- The description of the experimental protocol for the training phase is entirely missing. Which of the models were trained / fine-tuned? What was the training process? Did you fine-tune the foundation models? How was the dataset split / pre-processed? etc\n\n- 4.2.2, 4.3 are empty subsections. \n\n- The 3 pages filled with plots make the reading experience quite poor. Please, try to introduce a few plots at a time and discuss the related results close to the plots. \n\n- The choice of colors and line style in the plot in Figure 1 is quite unfortunate. The plot is very difficult to read.\n\nMinor comments:\n- There is a reference missing for the table in Appendix A.7.\n- Acronyms should be defined before usage (e.g., SDE, CDE...)\n- Malformed sentence at line 53", "broader_impact_concerns": "No concerns.", "additional_comments": "I would like to clarify that I am not an expert in the financial domain or the volatility forecasting task. Therefore, my analysis is limited to the machine learning contribution. I cannot evaluate whether the datasets used for the experiments are challenging enough or whether the modeling choices are sound, as they are strongly based on mechanistic models of volatility I do not know anything about.", "extra_scores": {}, "extra_text": {}}, {"review_id": "XFnHfmgCkP", "reviewer": "Reviewer_apP8", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "Summary Of Contributions:\n1. The paper presents a UDE model with small neural networks (small MLPs) for forecasting financial volatility that remains stable over multiple steps.\n2. The UDE keeps financial structure (mean reversion, volatility clustering, and jumps) and uses small neural networks to learn the related parameters (mean-reversion rates and long-run levels) so they keep clear economic meaning.\n\nStrengths:\n1. The authors prove the upper bound on the global error of the RK4 solver under their Lipschitz assumption.\nIt matters to prevent error accumulation over long steps that the error shrinks as h^4 (halving steps makes error drop by 2^4).\n2. The proposed UDE avoids using neural network to directly learn unconstrained drift which can explode and be unstable under long rollouts. The proposed UDE modulates a mean-reverting ODE with neural terms to preserve parameters having the economic meaning.\n\nWeaknesses:\n1. The paper intends to forecast financial volatility including jump discontinuities. However, the proposed \nUDE simulates the jump as a smooth, deterministic drift (λ(x) J(x)) inside ODE with fixed step RK4 (see Table 1 and A.1, A.2). This deterministic drift does not model real jump with Poisson arrival or jump size. It's better to clarify that handling jump discontinuities as average/approximate the jump effects within a continuous-time UDE.\n2. In abstract, mentioning jump discontinuities (discrete character) under continuous model framework needs clear clarification for the approximation effect.\n3. It's better to include comparision with jump diffusion SDE with Poisson jumps baseline in section 4 Experiments.\n4. The experimental design has a defect: 4.1 Asset Classes mentioned that performance on different assets bonds, crypto, equity during the year of 2008, 2020, 2022 under crisis are evaluated. However, 4.2 Results Table 2 only gives aggregated results. In volatility forecasting, asset class (bonds/equities/crypto) and market regime (calm/stress) make behavior differ substantially. The aggregated Normalized RMSE in Table 2 hides the heterogeneity. Please give per-asset and per-regime evaluation in addition to the aggregate result.\n5. The formatting needs to be improved. For example, a large blank space on page 7, a ‘?’ appears in citation [13], and ‘Table ??’ appears on line 500.", "claims_and_evidence_yn": "No", "claims_explanation": "1. The experimental design has a defect: 4.1 Asset Classes mentioned that performance on different assets bonds, crypto, equity during the year of 2008, 2020, 2022 under crisis are evaluated. However, 4.2 Results Table 2 only gives aggregated results. In volatility forecasting, asset class (bonds/equities/crypto) and market regime (calm/stress) make behavior differ substantially. The aggregated Normalized RMSE in Table 2 hides the heterogeneity. Please give per-asset and per-regime evaluation in addition to the aggregate result. (Also noted in the Weaknesses above.)\n\n2. In Experiments, the paper uses NRMSE, sMAPE, and R^2 as metrics. But in practice, MSE and QLIKE loss functions are commonly used on Realized Variance. Especially, QLIKE which has a better penalty design to handle noise in RV is recommended for volatility research. For example, the following classic paper \"Volatility Forecast Comparison Using Imperfect Volatility Proxies\" \n(https://public.econ.duke.edu/~ap172/Patton_robust_forecast_eval_11dec08.pdf) mentioned \"squared- error (MSE) and the QLIKE loss functions, two of the most widely-used in the volatility forecasting\". It's better to include MSE and QLIKE in section 4 Experiments and confirm the conclusion remain unchanged under MSE and QLIKE.\n\n3. EGARCH is a commonly used model. EGARCH is mentioned in A.8 Econometric Baselines \"GARCH-family models (GARCH, EGARCH, FIGARCH) were estimated\", but numerical result is missing for EGARCH.\n\n4. It's mentioned that \"Table 2 shows aggregated results across horizons 1, 5, 10, and 20 steps ahead\". However, in practice, risk managers often care about monthly or quarterly horizons, so 30, 60 and 90 steps should be included in the results.", "audience_yn": "Yes", "audience_explanation": "This paper will interest audiences in time-series forecasting, machine learning and differential-equation modeling, and financial risk modeling.", "requested_changes": "Please address the requested changes listed under Weaknesses, as well as the issues identified in the section explaining why the submission is not supported by accurate, convincing, and clear evidence.", "broader_impact_concerns": "I do not see any broader ethical concerns as the work focuses on volatility prediction and does not directly involve sensitive data and user privacy.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)