LETS Forecast: Learning Embedology for Time Series Forecasting
Authors: Abrar Majeedi
Viswanatha Reddy Gajjala
Satya Sai Srinath Namburi GNVV
Nada Magdi Elkordi
Yin Li
Venue: ICML 2025 poster (2025)
Keywords: Time series forecasting
dynamical modeling
Primary area: deep_learning->sequential_models_time_series
TL;DR: DeepEDM integrates nonlinear dynamical systems modeling with deep learning by using time-delayed embeddings and kernel regression, significantly improving time series forecasting accuracy over state-of-the-art methods.

Abstract: Real-world time series are often governed by complex nonlinear dynamics. Understanding these underlying dynamics is crucial for precise future prediction. While deep learning has achieved major success in time series forecasting, many existing approaches do not explicitly model the dynamics. To bridge this gap, we introduce DeepEDM, a framework that integrates nonlinear dynamical systems modeling with deep neural networks. Inspired by empirical dynamic modeling (EDM) and rooted in Takens' theorem, DeepEDM presents a novel deep model that learns a latent space from time-delayed embeddings, and employs kernel regression to approximate the underlying dynamics, while leveraging efficient implementation of softmax attention and allowing for accurate prediction of future time steps. To evaluate our method, we conduct comprehensive experiments on synthetic data of nonlinear dynamical systems as well as real-world time series across domains. Our results show that DeepEDM is robust to input noise, and outperforms state-of-the-art methods in forecasting accuracy. Our code is available at: https://abrarmajeedi.github.io/deep_edm.

LETS Forecast:
Learning Embedology for Time Series Forecasting
Abrar Majeedi Viswanatha Reddy Gajjala 12 Satya Sai Srinath Namburi GNVV2 Nada Magdi Elkordi Yin Li 12
Abstract
treat time series data as abstract patterns, and fall short in considering the underlying processes that generate them. Addressing this blind spot of deep models is important, be- cause at its core, time series data is not merely sequences of numbers; rather, these data represent the dynamic behavior of complex systems, encoding the interplay of various fac- tors over time. Indeed, many real-world time series data can treated as manifestations of time-variant dynamies (Brun- ton et al., 2022). Therefore, understanding the underlying systems can unlock more effective forecasting strategies.
Real-world time series are often governed by com- plex nonlinear dynamics. Understanding these un- derlying dynamics is crucial for precise future prediction. While deep learning has achieved major success in time series forecasting, many existing approaches do not explicitly model the dynamies. To bridge this gap, we introduce DeepEDM, a framework that integrates nonlin- car dynamical systems modeling with deep neu- ral networks. Inspired by empirical dynamic modeling (EDM) and rooted in Takens' theorem. DeepEDM presents a novel deep model that learns latent space from time-delayed embeddings, and employs kernel regression to approximate the underlying dynamics, while leveraging effi- cient implementation of softmax attention and allowing for accurate prediction of future time steps. To evaluate our method, we conduct com- prehensive experiments on synthetic data of non- linear dynamical systems as well real-world time series across domains. Our results show that DeepEDM is robust to input noise, and out- performs state-of the-ttr- methods in forecasting accuracy. Our code is available at: https:/ peearmajeedi github..ttt... b/deep_edm
Dynamical systems modeling characterizes the evolution of deterministic or stochastic processes governed by under- lying dynamics, thereby offering an appealing solution for time series forecasting. However, if system is not speci- fied, forecasting requires solving the challenging problem of inferring the underlying dynamics from observations. To address this, Empirical Dynamical Modeling (EDM) (Sugi- hara & May, 1990), data-driven approach built on Takens' theorem (Takens, 1981b; Sauer et al., 1991), was developed to recover nonlinear system dynamics from partial observa- tions of states. EDM leverages time-delayed embeddings to topologically reconstruct the system's 's state space from observed time series, which can then be used for forecasting. While EDM has demonstrated success in real-world appli- cations (Ye et al., 2015; Sugihara et al., 2012), it assumes noise-free data, requires separate modeling for individual sequences, and imposes constraints over its forecasting hori- zon, significantly limiting its broader practical applicability.
1. Introduction
bridge the gap, we propose a novel framework- DeepEDM that integrates EDM and deep learning, address- ing EDM's key limitations and introducing a new family of deep models for time series forecasting. Specifically, DeepEDM constructs time-delayed version of the input sequence, and projects them into a learned latent space that is more robust to noise. further employs kernel re- gression implemented using highly efficient softmax atten- tion (Vaswani et al., 2017), followed by a learned decoder, to model the latent dynamics and predict future values. Im- portantly, DeepEDM is fully differentiable, and thus can be learned end-to-end from large-scale data.
Time series forecasting fundamental across multiple do- mains including economics, energy, transportation, and meteorology, where accurate predictions of future events guide critical decision-making. Deep learning has recently emerged as the dominant approach, driven by its ability to leverage large datasets and capture intricate nonlinearity. While deep models excel in prediction accuracy, they often
'Department of Biostatistics and Medical Informatics, Univer- sity of Wisconsin-Madde 'Department of Computer Sciences, University of Wisconsin-Madison. Correspondence to: Yin Li syin.ii@wisc.euu>.
Proceedings of the 42nd International Conference Machine Learning, Vancouver, Canada. PMLR 267, 2025. Copyright 2025 by the author(s).
DeepEDM connects traditional EDM and modern deep learning. On one hand, it significantly extends EDM by im-

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LETS Forecast: Learning Embedology for Time Series Forecasting
proving robustness against measurement noise, enabling the learning of single parametric model generalize across sequences, and supporting longer forecasting horizons. On the other hand, it integrates the rigor of dynamical systems modeling with the flexibility and scalability of deep learn- ing, leading to a variant of Transformer model for time series forecasting, and providing theoretic insights for other Transformer-bawee models (Liu et al.. 2024a; Nie et al., 2023; Chen et al., 2025).
channel-wise attention framework, have further improved the forecasting performance of attention-based models.
2.2. Learning Dynamical Systems for Forecasting
Learning dynamical systems for time series forecasting has garnered considerable interest within the research commu- nity. Many prior works builds on Koopman's 's theory (Mezić, 2021; Brunton et al., 2022), which represents nonlinear system with a linear operator in an infinite-dimensionaal space. Examples includes Koopman Autoencoder (Lusch et al., 2018; Takeishi et al., 2017) and K-Forecast (Lange et al., 2021). Both approximate the Koopman operator in high-dimensional space to effectively model nonlinear dynamics. These approaches enable scalable forecasting for complex systems by simultaneously learning the measure- ment function and the Koopman operator.
Our main contributions are thus three folds. First, we pro- pose DeepEDM, a novel framework inspired by dynamical systems modeling that leverages demeyee embeddings for time series forecasting. Second, DeepEDM, grounded in Takens' theorem, addresses key limitations of EDM, and sheds light on prior Transformer-basee time series models. Third, extensive experiments synthetic datasets and real- world benchmarks, demonstrate of-eeofft forecasting performance of DeepEDM.
Recent developments, including Koopa (Liu et al., 2024b) and Deep Dynamic Mode Decomposition (Deep- DMD) (Alford-Lago et al., 2022). extend this frame- work. Koopa enhances the forecasting of nonlinear systems through modular Fourier filter combined with Koopman predictor. Together, these components hicrarchically disen- tangle and propagate time-invariant and time-variant dynam- ics. DeeDDMD employs deep learning to traditional DMD, facilitating the identification of coordinate transformations that lincarize nonlinear system dynamics, thus capturing complex, multiscale dynamics effectively. A more recent work, Attraos (Hu et al., 2024), has explored alternative perspectives through chaos theory and attractor dynamics.
2. Related Work
2.1. Deep Learning for Times Series Forecasting
There has been major progress in time series forecasting thanks to deep learning. Early approaches predominantly consider Recurrent Neural Networks (RNNs), especially Long Short-Term Memory (LSTM) networks (Hochreiter & Schmidhuber, 1997; Yu et al., 2017), which are adept at cap- turing long-term dependencies. Subsequent developments, such as LSTNet (Lai et al., 2018) and DeepAR (Salinas et al., 2020), integrate recurrent and convolutional struc- tures to enhance forecasting accuracy. Temporal Convo- lutional Networks (TCNs) (Bai et al., 2018), and methods like MICN (Wang et al., 2023a) and TimesNet (Wu et al., 2023), leverage multi-scale information and adaptive re- ceptive fields, improving multi-horizon forecasting capa- bilities. Recent works find that Multi-Layer Perceptrons (MLPs) can achieve competitive performance. Notably, TimeMixer (Wang et al., 2024a) presents a sophisticated ML.--based architecture that incorporates multi-scale mix- ing, outperforming previous MLP models such as DLin- ear (Zeng et al., 2023) and RLinear (Li et al., 2023).
2.3. Empirical Dynamical Modeling
EDM (Chang et al., 2017: Sugihara & May, 1990) presents an alternative approach to model nonlinear dynamics. Rooted in Takens' theorem (Takens, 1981b). it relies on delay-coordinate embeddings to reconstruct the underly- ing attractor, thereby preserving the essential topological properties of the original dynamical system. Unlike Koop- man, which linearizes nonlinear dynamics in a carefully chosen high dimensional space (approximation to an infinite- dimensional space), EDM can topologically reconstruct sys- tem dynamics using low dimensional observations, or even with a scalar observation at each time step (Takens, 198 bb; Sauer et al., 1991). EDM is thus particularly attractive for problems with limited observation of the system states.
Transformer-based (Vaswani et al., 2017) models have shown to be highly effective for long-term forecasting (Chen et al., 2025). Architectures like Reformer (Kitaev et al., 2020), Pyraformer (Liu et al.. 2021), Autoformer (Wu et al., 2021), and Informer (Zhou et al., 2021) have enhanced the scalability and efficiency of attention mechanisms, adapt- ing them for longer range time series forecasting. Sub- sequent innovations such as PatchTST (Nie et al., 2023), which proposes a patching- proach along with instance normalization (Ulyanov et al., 2016), and Trransformer (Liu al., 2024a), which utilizes
A line of prior research (Mezié & Banaszuk, 2004; Arbabi & Mezić, 2017; Mezić, 2022) has also explored the connec- tion between Takens' theory and Koopman operator theory. Hankel-DMD (Arbabi & Mezić, 2017) applies dynamic mode decomposition to time-delayed time series, effectively performing Koopman spectral analysis on Takens' embed- dings. Numerical issues associated with Hankel-DMe have been discussed in (Mezić, 2022).
channel independent ap-
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LETS Forecast:
Learning Embedology for Time Series Forecasting
2.4. Chaotic Time Series Forecasting
are constrained, the dynamics can be topologically recon- structed, perhaps surprisingly, even with univariate measure- ments F1at. With slight abuse of notations, we now restate Takens' theorem in our setting.
A related research direction focuses on forecasting chaotic time series via state space reconstruction, mirroring the un- derlying principles of EDM. Pioneering work by (Farmer & Sidorowich, 1987) introduced local approximation tech- niques within reconstructed state spaces using delay em- beddings for short-term predictions. Subsequent stud- ies explored feedforward neural networks for learning di- rect mappings from reconstructed phase states to future states (Karunasinghe & Liong, 2006). Recurrent models, especially Echo State Networks (ESNs) (Jaeger & Haas, 2004), and variants like robust ESNs (Li et al., 2012), fur- ther improved resilience to noise and outliers.
Theorem 3.1. (Takens, 198la) Let M be a compact mani- fold of dimension d defining the space of states x and assume the observed time series data is univariate, i.e., y E R. For pairs of dynamics o and observation function h, where 4 .M + MM is a C² smooth diffeomorphism, i.e., &b must be bijective and both & and its inverse & are C² smooth, and h : :M + I. is a C2 smooth function, it is a generic property that H( h) : .M- + B2d-1 defined by (h(x)), h (x)())), (a()((x)) ((.,), h h(()2(x)) (x)) is an immer- sion, i.e. H is injective and both H. and H are differen- tiable.
3. DeepEDM for Time Series Forecasting
We consider time series generated by discrete-time nonlin- dynamical systems, though all derivations can be readily extended to continuous-time systems. A discrete-time non- linear dynamical system is defined as a recurrence relation in which nonlinear function & governs the evolution of the state variables Z2 E Rd at time step t:
The (2d+1)-D vectors (h((), thus preserve the topology of the states 24. By setting x - 24-2a, it is easy to note that these vectors are [1yt22 31-2441...... 3e]3, i.e., time-delayed version of the observed time series. The theorem thus states that given time series of ID measurement yt, its time delayed ver- sion d1:t1 D. = (Ut-2a: Vt Vt-2d+1.... Br)) has similar topol- ogy with the states zo.t- Therefore, we can instead model the induced dynamics of fu:: to recover properties of the underlying dynamics of Ti as illustrated in Figure ((a).
(())))))))
F++1 (x.).
(I)
Oftentimes, the states of the system He can not be directly
observed and the governing equation & is unknown. Instead, common assumption is that measurements y of the states I, can be acquired using
It worth noting that Takens' theorem has two restrictive as- sumptions: (1) the state space must be compact manifold; and (2) measurements are univariate. Recent developments have extended the theorem to more general settings, account- ing for the state space as a compact invariant set within finite Euclidean space (Sauer et al., 1991), or the measurements as multivariate vectors (Deyle & Sugihara, 2011).
y h(x))) +
(2)
where h is an unknown measurement function that maps a system state Ie to its observation ye with a time-invariant stochastic noise 6.
Our goal is time series forecasting, i.e., predicting future observations yT+1:T+ based existing ones f/:r7- Virr often referred to as the lookbuck window with length T, and yT+11T+ as predictions with its forecasting horizon H. Without knowing the governing equation T or the measure- ment function h, this forecasting problem is very challeng- ing even with a small amount of noise e. In what follows, we introduce the theoretic background, present our approach, and describe its practical instantiation.
Empirical Dynamic Modeling (EDM)
Built on Takens' theorem, EDM (Sugihara & May, 1990;
Dixon et al., 1999; Sugihara et al., 2012; Chang et al., 2017) provides a computational method to reconstruct a system's 's state space from time series of its univariate measurements. We now briefly describe EDM with Simplex projection, which lays the foundation for our approach.
Simplex projection assumes univariate measurements yırr and considers its time-delayed version Ộ1. with dt E R24+1, i.e., time-delayed by 2d + 1 steps. To forecast a future time step yr+ (A/ 2 1), it first finds 2d + 2 near- est neighbors (6x,) J. E [L,..., 2d + 2] for dr using a pre- specific similarity metric, i.e., a kernel function k(g, 1/)', These nearest neighbors {ØN,] are assumed to define a sim- plex in the 1d+1-D space of ů, i.e., geometric structure that generalizes a triangle (in 2D) or a tetrahedron (in 3D)
3.1. Preliminaries: Takens' Theorem and EDM
Takens' Theorem
Takens' theorem establishes the feasibility to "recover" the underlying dynamics defined by & without knowing the observation function h and assuming zero noise (i.c., e = 0). In this case, forecasting becomes straightforward and only involves forwarding the uncovered dynamics. Intuitively, the theorem states that if Φ, h and the state space of x
'The original method in (Sugihara & May, 1990) used the radial basis function kernel k(j. 8') = exp( -lpi - 0 0/11"/20²)
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LETS Forecast: Learning Embedology for Time Series Forecasting
[e]
[b]
Hidden Dynamics
Reconstructed Dynamics
wmm.m +
Noise
Identify saft- Neerest Neighbors
Observed Data (Noisy)
Latent Dynamics (Noise-free)
Forecast via Kernel Regression
Projection
m M
/ M
y
Z
Decoder
~
Time Delay
M
6x-;,))
Observed Data(y)
Delayed
Time Delayed
Latent Projection
Forecast (yPrec)
Embeddings (9)
Embeddings (9)
Figure 1. (a) Takens' theorem in action. The state space of an unknown nonlincar dynamical system is reconstructed using time-delayed embeddings from observed time series measurements (noise free). (b) Overview DeepEDM. Time-delayed embeddings are constructed to model the system's 's underlying state space. These embeddings are then mapped into a learned latent space that is robust to measurement noise. Forecasting is performed via kernel regression followed by a learned decoder, where soft nearest neighbors for regression are defined in the latent space. This model, resembling the key idea of EDM, is fully differentially and thus can be learned from end-to-end.
to arbitrary dimensions. yr- is then predicted by linear re- re-weighting on this simplex, given by
erates initial predictions, relaxing the constraint on forecast- ing horizon; (2) a learned encoder embed time-delayed time series into a latent space, gaining robustness against input noise; (3) a kernel regression to predict future data in the latent space, re-assembling Simplex projection while al- lowing for efficient and differentiable implementation; and (4) a decoder to output the final predictions and mitigate noise. Collectively. DeepEDM is fully differentiable and enables end-to-end learning of a single parametric model for forecasting that generalizes across time series, avoiding per-sequence modeling in EDM.
AT
2242
WE Σ we UN,+AG
(3)
where the weight is given by wy le (ÔT, 2N.) ). Again, Na indexes the 2d + 2 nearest neighbors of yt, and yN denotes the observed data At steps after N₁. We note that Equation (3) can be viewed as the Nadaraya-Watte Watson esti- mator using 2d + 2 nearest neighbors, where the regressor connects the input of IN, to its output of UN.+AI-
To simplify our notations, we describe DeepEDM in the con- text of univariate time series forecasting in this section. For multivariate time series, DeepEDM applied channel-wise meaning single DeepEDM model shared across individ- ual variates strategy widely used in prior works (Nie et al., 2023; Zeng et al., 2023). We now introduce individual components, present the training scheme, and discuss links to Transformer- models.
Simplex projection can be also considered locally linear approximation to the manifold of the time-delayed observa- tions v. which is topologically equivalent to the state space of x. The key assumption is that f1.T sufficiently covers the manifold, such that the nearest neighbors (ŪN,) } of ýr correspond to underlying states similar to 2,, This assump- tion allows an empirical approximation of forwarding & for forecasting, using the the future data of these nearest neighbors (yN,+dr)- However, it also imposes a practical constraint: the forecasting horizon (H) must be significantly shorter than the length of the lookback window (T)).
Modeling
Initial prediction. DeepEDM starts with a simple, base prediction model f() (e.g.. a linear model an MLP). f()) takes the input of the lookback window 91.r with y E R (univariate single channel in a multivariate time series), and outputs the predictions y" for H steps
3.2. Our Approach: DeepEDM
Despite its success (Ye et al., 2015; Sugihara et al., 2012), EDM with Simplex projection has three key limitations. First, it assumes noise-free measurements, leading to sig- nificant performance degradation when forecasting in the presence of noise. Second, it models each sequence inde- pendently, disregarding patterns shared across time series. Third, it imposes constraint that the forecasting horizon must be much shorter than the lookback window.
VT+ITT+ = f(n:r).
(4)
This initial prediction allows us to concatenate the lookback window 91.7 and the predicted window + form- ing a new time series [yI.T, 97+ DeepEDM will now operate this extended sequence and further refine the initial prediction, bypassing EDM's constraint on the forecasting horizon. This is particularly helpful for long- term forecasting, where T might be smaller than H. Time delay and encoding. DeepEDM further time-delays the ex- tended sequence [y1:T, + d) and considers learned
To address these limitations, we present DeepEDM, a novel deep model that builds on the key idea of EDM, leveraging strengths from both paradigms. DccpEDM, as shown in Figure 1(b), consist of (1) a base forecasting model that gen-

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LETS Forecast: Learning Embedology for Time Series Forecasting
encoder Enc(-) project the time-delayed signals into latent space. Formally, this is given by
C C= A lly mr +(1
Vy med
(8)
iT+H = ((mar - 21:T+H Enc (fur+H),
Can
Cat
(5)
where V denotes the first order finite difference and À is the balancing coefficient. Namely, our loss minimizes the Lp norm of the prediction errors (Cen) and its temporal differences (Lu). We also find it helpful to consider an adaptive À following (Xiong et al., 2024), especially for long-term forecasting problems. Further details of our loss function can be found in the Appendix A.2.
where 6r) denotes time delay operator with 8y delay steps-a hyperparameter of DeepEDM. ÿr is thus the time- delayed embedding of the concatenated sequence. Note that zero padding is added before the sequence to preserve the temporal dimension. The encoder Enc()) is realized using a neural network with learnable parameters. Enc(-) is de- signed extract features from the time-delayed embeddings of an input sequence, enabling meaningful comparisons among these embeddings with noisy measurements.
Discussion
Relationships to Transformer-based models. DeepEDM shares a strong conceptual connection with Transformer models widely used in time series forecasting. Specifi- cally, the notion of time-delay embedding in EDM and DeepEDM can be viewed as a special case of local win- dow patching (Nie et al., 2023). Moreover, the combination of encoder, kernel regression, and decoder resembles the structure of Transformer block with self-attention (Chen et al., 2025), albeit with distinct definitions of queries, keys, and values. From this perspective, DeepEDM can be in- terpreted Transformer-like model with input patching, which refines initial predictions from simple base model. Indeed, patching, Transformer architectures, and cascaded prediction have all been proven to be highly effective for time series forecasting.
Simplex projection with kernel regression. DeepEDM further employs kernel regression for prediction, extending the key idea of Simplex projection in EDM. While Simplex projection finds ((= 5 - nearest neighbors- - an opera- tion that is not differentiable, we propose to instead leverage all data points, again using the Wadaraya-Watson estimator. In this case, we rely on the choice of the kernel k(:--) to down-weight irrelevant data. Formally, this is expressed as
Σ z)) Betar
11 Etm k(2,, ze)
(6)
where 1' E [T, H At) and we simply set At= 1 for a single-step forward prediction. We choose k(z,, 2x)) = exp((x,, 2/)/T) with J to control its decay, and leverage highly optimized softmax attention for efficient implemen- tation (with 7 as the temperature in softmax). While 7 can be learned from data, we empirically find that doing has minimal impact on overall performance, and keep 1.
3.3. Model Instantiation
Base prediction model, encoder, and decoder. The base predictor f(-) is realized using multilayer perceptron (MLP) shared across all variates. Given historical input y E R , f(-) maps d-th variate's time series ya RTT to prediction 2a E R., yielding the initial prediction y" E DDH Subsequently, the lookback y and the initial forecast y" are concatenated and time delayed by by steps, resulting in hu.T E IR Dx6r ((T+H).
Notably, unlike Simplex projection (Eq. 3), which predicts a scalar corresponding to a single step in a univariant time series, our kernel regression (Eq. 6) predicts a vector of size 57 representing the time-delayed version of the time series.
Prediction decoding. Finally, DeepEDM decodes the out- prdd
put UT+IT++ based on UT+11++ using a decoder Dec(-)
The encoder Enc(-) instantiated as single linear operator shared across all variates. It operates on the time-delayed se- quence Ū1:T+ H R DDx6 where each delay vector of dimension 8T is linearly projected to learnable latent with dimension M >> 8r, resulting in the embeddings 21:T- E DD This lightweight Enc(.) seeks to preserve the local geometry of the time-delay embed- ding while enabling expressive comparisons in the latent space used for kernel regression (Eq. 6), which generates the time-delayed prediction Ūr+1:T+H-
ped o # H = Dec (#r-1:44))
(7)
where Dec(-) is realized with a lightweight neural network with learnable parameters. Dec(-) Iearns to reconstruct the predicted time series from its time-delayed version and, crucially, denoises the output to mitigate the effects of mea- surement noise introduced during kernel regression.
Training
DeepEDM includes learnable parameters in the base model f()), the encoder Enc(-), and the decoder (-c(-). Our learn- ing objective is to jointly optimize these parameters to min- imize prediction errors on the training set. Omitting sub- scripts for simplicity, our training loss is defined as:
The decoder Dec(-) maps the time-delayed forecast ÜT+1 E I MX H back to the original time series space. It is implemented using a lightweight MLP shared across all channels. For each channel, the (M x H) latent matrix is first flattened into a vector, which is then passed
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LETS Forecast: Learning Embedology for Time Series Forecasting
yorrd (Forecast)
Noise
Medium Noise
High Noise
6.0
Simplex
IED Block
Gate
Keepa
Mhansforer
Ours
Decoder
0.5
|| Freuren
13
15
15
Σ Y W
MSE
W
xN
Prediction Length
Prediction Length Prediction Length
Regression: on: Wij (22..7)
Figure 3. Results with synthetic data from Lorenz systems. We plot MSE under varying prediction lengths on non-Chaotic (top) and chaotic (hottom) Lorenz. DeepEDM significantly outperforms baselines in both chaotic and non- non-chaotic regimes.
Encoder
domains, including weather, electricity, traffic, and finance. Finally, we provide extensive analysis to assess DeepEDM's ability to generalize to unseen time series, and to study its key design choices. Due to space limits, part of our re- sults, along with extended benchmarks and visualization, are provided in the Appendix.
9uT
9T+LT+N
Time Delay (+)
Lookback
Initial Forecast
Figure 2. DeepEDM block can be stacked, with cach subsequent block iteratively refines the prediction of the previous one.
4.1. Experiments on Synthetic Data
through the MLP to produce the final forecast in R. yield- ing the output yood E DDx (ec(-) aims to reconstruct the forecast using its noisy time-delayed version.
We evaluate DeepEDM on synthetic time series generated from (1) non-chaotic Lorenz (Lorenz, 1963), (2) chaotic Lorenz (Lorenz, 1963), and (3) chaotic Rössler (Rössler, 1976) systems. Lorenz and Rössler systems are widely used to study chaotic and non-chaotie dynamics.
DeepEDM block. We combine the time delay operation, encoder, kernel regression, and decoder into DeepEDM block, as shown in Figure 2. This block receives the his- torical input E Dx1' in tandem with an initial forecast y E I Dx produced by the base predictor, and predicts fu- ture time series yood R Dx... Importantly, the DeepEDM block is stackable: the output of one block serves as the input forecast to the next, enabling the model to iteratively refine its predictions. To improve gradient flow and training stability, we introduce skip connections from the initial fore- cast yd to the final output y modulated by learnable gating function implemented as a simple linear layer.
Simulation, setup, and baselines. To simulate noisy data, we inject Gaussian noise NO, o of various magni- tudes of anoise E {0.0, 0.3, 1.0, 1.5, 2.0, 2.5} to the 3 afore- mentioned systems, resulting in total of 18 synthetic datasets (see Appendix A.7). We benchmark DeepEDM against three baselines, including EDM with Simplex, Koopa (a deep model integrating Koopman theory), and caansformrr-bbseee Since Simplex is inher- ently a univariate forecasting method, we run it indepen- dently on each variate and aggregate the results to obtain multivariate forecasts. The performance is reported by mean squared error (MSE) and mean absolute error (MAE).
iTTrasformer
Our full model. Our DeepEDM model consists of base predictor f(-), followed by several stacked DeepEDM blocks. Given lookback window y. the base predictor generates coarse forecast y², which is successively refined through stacked DeepEDM blocks. All components are differentiable and jointly trained with our loss in Eq. 8.
Results. Figure 3 shows the forecasting results of all meth- ods across noise levels and prediction horizons. In the low- noise and non-chaotic settings, all methods exhibit compa- rable performance. However, as noise increases, EDM with Simplex degrades sharply, while DeepEDM remains robust, achieving lower MSE across all conditions. It also outper- forms Koopa and iTransformer by a small but meaningful margin (see Table 11 in Appendix). In chaotic regimes, DeepEDM's advantage is more pronounced, consistently outperforming Simplex at all horizons and surpassing Koopa and iTransformer for longer forecasts. These results under- score DeepEDM's 's robustness in noisy, chaotic environments while exceeding both classical EDM and modern baselines. Additional results, including those for the Rössler system,
4. Experiments and Results
We evaluate DeepEDM across a wide range of synthetic and real-world benchmarks. Our initial evaluations leverage synthetic datasets derived from well-established nonlinear dynamical systems, allowing us to systematically analyze DeepEDM's 's capacity to capture complex temporal depen- dencies. Further, we compare DeepEDM against of- tre-art deep models on real-world datasets spanning diverse
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LETS Forecast: Learning Embedology for Time Series Forecasting
Table 1. Multivariate forecasting results with different forecast lengths H E (24, 36, 48, 60) for IIII and H E {48, 96, 144, 192] for others. We set the lookback length T = 2H. Bold indicates the best performance, while 2nd best is underlined. In case of a draw, both models are considered winners. Gray represents dynamical systems. Source: When available, results are taken directly from (Liu et al.,
2024b); otherwise reproduced using their official code run with reported metrics averaged over 5 runs with different random seeds.
Models
Ours
Koopa
KNF
Attrans
CycleNet
Transformer PaichTST TineMixer
DLinear
FITS
MICN
Naive
Metric
MSE
MAE
MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE MSE MAE
MSE MAE
48
0.324
0.357
0.336 0.377 0.876 0.709 0.341 0.371 0.331 0.370 0,343 0,380 0.337 0.375 0.336 0.375 0,343 0,571 0,344 0.370 0.375 0.406 1.268 0.695
ETTh
96
0.365
0.384
0.371 0.405 0.975 0.744 0,387 0.402 0.389 0.404 0.392 0.411 0,372 0.393 0.388 0.405 0.379 0.393 0.381 0.395 0,406 0.429 1.294 0.713
0.388
0.398
0.405 0.418 0.801 0.662 0,415 0.422 0.415 0.422 0.424 0.430 0.394 0.412 0.413 0.421 0.393 0.403 0.396 0.406 0.437 0.448 1,316 0.725
192
0.407
0.421
0,416
0.429 0.941 0.744 0.429 0.434 0.433 0.436 0.446 0.449 0,416 0.439 0.443 0.447 0.407 0.416 0.405
0:414
0,518 0.496 1,325 0.733
48
0.225
0.288
0.226 0.300 0.385 0,376 0,230 0.301 0.238 0.305 0.243 0.314 0.223 0.297 0.230 0.302 0,226 0.305 0.227 0.298 0.260 0.336 0.344 0.374
B1
96
0.289
0.333
0.297 0.349 0.433 0.446 0,302 0.350 0.306 0.357 0,302 0.356 0,300 0.353 0.298 0,350 0.294 0,351 0.287 0.341 0.343 0.393 0.432 0.422
144
0.324
0.362
0.333
0.381 0,441 0,456
0.355 0.383 0.350 0.388 0,346 0.386 0.346 0.390 0.339 0.383 0,354 0,397 0.315 0.363 0.374 0,411 0,484 0.448
1920
0.351
0.377
0.356 0.393 0.528 0.503 0.373 0.399 0.377 0.407 0.383 0.409 0.383 0.406 0.359 0.406 0.385 0.418 0.334 0.376 0.455 0,464 0.534 0.472
48
0.277
0.318
0.283
0.333
1.026
0.792 0.312 0.355 0.283 0.336 0.314 0.358 0.286 0.336 0.302 0.349 0,322 0,355 0.324 0.357 0.294 0.353 1.165 0.638
ETTml
96
0.288
0.328
0.294
0 0.345 0.957 0.782 0.314 0.355 0.302 0.353 0.304 0.354 0,299 0.346 0.299 0.348 0.309 0.346 0.310 0.346 0.306 0.364 1.214 0.665
14400
0.308
0.344
0.322
0.366 0.921 0,760 0,332 0.368 0.327 0.368 0.331 0.373
0,325 0,363 0.326 0.365 0.327 0.359 0.326 0.358 0.342 0.390 1.246 0.682
192
0.322
0.353
0.337
0.378 0.896
0.731 0.349 0.378 0.346 0.382 0.345
0.383
0.343 0.375 0.345 0.378 0.337 0.365 0.338 0.365 0.386
0.415 1,261 0.690
48
0.133
0.221
0.134
0.226
0.621 0.623 0.139 0.236 0.123 0.216 0.139 0.234
0.135 0.231 0.136 0.229 0,144 0.240 0.145 0.242 0.131 0.238 0.220 0.295
Er/m2
96
0.169
0.248
0.171 0.254
1,535
1.012 0.174 0.259 0.164 0.249 0.181 0.269
0.171 0.255 0.174 0.257 0.172 0.256 0.172 0.257 0.197 0,295 0,267 0.328
144
0,203
0.271
0.206
0.280
1.337
0,876
0.209 0.284
0.212 0.286 0,214 0.294
0.205
0.282 0.207 0.284 0.200 0.276 0.200 0.277 0.210 0.297 0,307 0,352
192
0.224
0.289
0.226
0.298
1.355
0.908
0.233 0.302
0.231 0.302 0.238
0.310
0.221 0.294 0.229
0.297 0.219 0.290 0.220 0.291 0.248 0.328 0.340 0.371
48
0,161
0.247
0.130 0.234
0.175 0.265
0.192 0.268
0.120
0.215 0.134
0.226
0.147
0.246 0.142 0.235 0,158 0,241 0.203 0.279 0.156 0.271 1,543 0.925
ECL
96
0.137
0.232
0.136 0.236 0.198 0.284 0.150 0.244 0.127 0.222 0.134 0.230
0.143 0.241 0.134 0.227 0.153 0.245 0.154 0.248 0.165 0.277 1.588 0,946
0.145
0.239
0.149 0.247 0.204
0,297 0.151 0.246
0.138 0.232
0,146 0.240
0.145
0.241
1..45 0.235 0,152 0.245 0.152 0.246 0.163 0.274 1.605 0.953
192
0.151
0.244
0.156 0.254 0,245 0.321 0.154 0.249
0,146 0.241
0.155
0.249
0.147 0.240
0. 166 0.255 0.153 0.246 0.154 0.247 0.171 0.284 1.596 0.951
Esehunge
48
0.042
0.142 0.042
0.143 0.128 0,271 0.045
0.147 0.044 0,144 0.045
0.148
0.044
0.144 0.043 0.143 0.043 0.145 0.054 0.180 0.117 0.248 0.042 0.139
96
0.088
0.205
0.083 0.207 0,294 0,394 0.093 0.213 0.089 0.209 0.095 0.219 0.085 0.204 0.084 0.203 0.084 0.220 0.113 0.261 0.108 0.251 0.081 0.196
144
0,133
0.255
0.130 0.261 0.597 0.578
151 0.274 0. 144 0.267 0.154 0.283
0.1 0.260 O. 144 0.270 1.1 0.253 133 0.258 0.152 0.301 0.122 0.244
192
0.178
0.301
0. 184 0.309 0.654 0.595 0,205 0.323 0.207 0.322 0,212 0.334
0.174 0,300 0. 196 0.316 0.178 0.299 182 0.305 0.187 0.331 0.167 0.289
48
0,448
0,286
0.415 0.274 0.621 0.382 0.612 0.396 0.437 0.290 0.369
0.257 0.426
0.286 0.445 0.283 0.488 0,352 0.704 0.419 0,496 0,301 2.641 1.057
Traffie
96
0.383
0.259
0.401 0.275 0 0.645 0.376 0.439 0,300 0.406 0.276 0.365
0.259 0,413 0.283 0.406 0.277 0.485 0,336 0.457 0.306 0.511 0.312 2.715 1.077
0.380
0.258
0.397 0.276 0
0.402 0.423 0.294
0.402 0.275 0.373 0.266
0.405 0.278 0.391 0.263
0.452 0.317 0.432 0.293 0.498 0.309 2.739 1.084
192
0.387
0.262
0.403
0.284 0.699 0. 0.405 0.421 0.295 0.402 0.275 0.374
0.267 0.404 0.277 0.424 0.293 0.438 0,309 1,313 0.776 0.494 0,312 2.747 1.085
48
0.138
0.168
0.126
0.168 0.201 0,288 0. 0.149 0.1
0.129 0.171 0.137 0.174
0.140 0.1 0.131 0.174
0,156 0.198 0.157 0.200 0.157 0.217 0.194 0.193
Weather
96
0.157
0.192 0.154 0.205 0.295 0.308 0. 1116881 0.214
0.
15155 0 0,203 0.169 0.215 0.160 0.206 0. 0.1 0.205 0.186 0.229 0.187 0.231 0.187 0.250 0,259 0.254
144
0,174
0.210
0.172 0.225 0.394 0,401 0.184 0.231
0.171 0.223 0.187 0.234
0.174 0.221 0.173 0.223 0.199 0,244 0.199 0.244 0.197 0.257 0,284 0.274
192
0.191
0.226
0.193
0.241 0.462
0.437
0.202
0.249
0.192
0.243 0,206 0.253
0.195 0.243 0.193 0.243 0.217 0.261 0.217 0.261 0.214 0.270 0.309 0.292
24
1.799
0.797
1.621
0.800
3.722
1.432
2.188 0.940 1,966 0.888 2.063 0.881 2.147 0.899 2.624 1.118 3,311 1.311 4.380 1,558 6,213 1.622
36
1.655
0.768
1.803
0.855
3,941
1.448
2.113 0.949 1.827 0.865 2.178 0.943 1.892 0.894 2.693 1.156 3.112 1.232 3.314 1.313
7,714 1.906
48
1.616
0.789
1.768
0.903
3,287
1,377
2.437
1.084
1.849
0.919
1.748 0.908
1.916 0.896 1.874
0.915 2.852
1,229
3.156 1,290
2.457
1.085
7,851 1.952
60
1.719
0.831
1.743
0.891
2.974 1,301
2,341
1 1.064
1.872 0.932
2.077 0.999
1. 1.981 0.917 2. 287 0.991 2.554 1.144 3,337 1,280 2.379 1.040
6.885 1.788
1d Count
36
11
2
Note: The official code of Attraos (Hu al., 2024) does not support H = (24,36) for ILI datasct. Therefore, we report these entries as empty rather than extensively modifying their code to make it work.
are provided in Appendix A.7.
ETTm2 (Zhou al., 2021), National Illness (ILI) (Lai et al., 2018), Solar-Energy (Lai et al., 2018) (see ap- pendix), Electricity (see appendix), Traffic (PeMS) (Wu et 2021), Weather (Wetterstation) (Wu et al., 2021), and Exchange (Lai et al., 2018). For univariate forecasting, we leverage the well-established M4 dataset (Makridakis et al., 2020) (see appendix), which contains 6 subsets of period- ically collected univariate marketing data. These datasets encompass different domains and exhibit diverse temporal
4.2. Experiments on Forecasting Benchmarks
Moving forward, we conduct comprehensive evaluations on standard time series forecasting benchmarks.
We consider both multivariate and univariate time series forecasting. For multivariate forecasting, we eval- uate on 10 real-world datasets: ETThI, ETTh2, ETTml,
Datasets.
7

---

LETS Forecast: Learning Embedology for Time Series Forecasting
patterns, allowing for robust assessment.
Table 2. Generalization to unseen time series. Each model
trained on a subset of sequences and evaluated on disjoint, unseen
Setup. Our experimental protocol adheres to the pre-
sequences from the same dataset. DeepEDM achieves the hest MAE and MSE in 39 out of 48 settings.
processing methods and data split ratios established by
prominent prior works such as TimesNet (Wu et al., 2023) and Koopa (Liu et al., 2024b). For all experiments, we use the Time-Series-Libree Liibrar (Wang et al., 2024b) to en- sure consistency and comparability. For our main results, we adopt the adaptive lookback windowing approach from Koopa (Liu al., 2024b), where the lookback window length T is set to twice the forecast horizon H. We also report results with lookback window search in the appendix.
Models
Oura
Koopa
PaccISTT
iTrzoformee
48
0.2182
0.2980
0.2383
0.3190
0.2312
0.3050
0.2474
0.3190
96
0.2230
0.3120
0,2382
0.3270
0.2601
0.3370
0.2535
0.3340
0.2285
0.3190
0.2598
0.3420
0.2567
0.3380
0.2634
0.3420
192
0.2510
0.3400
0.2555
0.3420
0.2637
0.3430
0,2702
0,3480
48
0.0931
0.1850
0.1181
0.2160
0.1114
0.2090
0.1107
0,2110
ETT12
96
0.1377
0.2260
0.1517
0,2410
0.1653
0.2590
0.1555
0.2510
144
0.1795
0.3640
0.1979
0.2850
0.2265
0.2980
0.1914
0.2760
192
0.1956
0.2770
0.2115
0,2950
0.2297
0,3170
0,2209
0.2970
48
0.2068
0.2680
0.2377
0.2960
0.2208
0.2830
0.2305
0.2910
Baselines. We consider a set of strong baselines. While em- phasizing comparisons with dynamical system-based meth- ods such as Koopa (Liu et al., 2024b), Attraos (Hu et al., 2024) and KNF (Wang et al., 2023b), we also include other popular baselines. These include TimeMixer, FITS, and DLinear, as well as Transformer- based models such as iTransformer and PatchTST. Addi- tionally, we also benchmark the Natve baseline as described by (Hewamalage et al., 2023) (i.e. predicting the last value of lookback window as forecast) to provide the simplest benchmark to assess relative performance.
96
0.2141
0.2780
0.2337
0,3020
0.2090
0,2840
0.2386
0.3020
144
0.2142
0,2880
0.2518
0.3180
0,2134
0.2960
0,2590
0.3180
192
0.2194
0.2960
0,2454
0,3200
0.2212
0.3070
0,2512
0.3200
48
0.0544
0,1470
0.0641
0.1640
0,0710
0.1740
0.0892
0.1950
96
0.0689
0.1590
0.0814
0.1820
0.0768
0.1750
0.0861
0.1890
models like
144
0.0784
0,1710
0.0920
0.1910
0.0917
0.1880
0.1004
0.1990
0.1024
0.1073
0.2060
0.1115
0.1074
0.2070
Focheegs
48
0,0388
0.1290
0.0459
0.1420
0.0431
0.1370
0.0428
0.1390
0.0783
0.0936
0,2120
0,0828
0.0912
0.2030
144
0.1330
0.2390
0.1725
0.2820
0.1279
0.2490
0.1772
0,2800
0.1754
0.2706
0.3560
0.1667
0.2136
0.3140
48
0.2915
0.24/0
0,3179
0.2780
0,3030
0.2410
0.3967
0,2870
Wezthe
0.2977
0.2510
0.3142
0,2760
0.2934
0,4462
0.3260
144
0.29228
0.2520
0.3101
0.2820
0.2841
0,2570
0,4085
0.3220
0.2917
0.2630
0.3261
0.2900
0.2907
0.2620
0,4393
0.3430
Results. Our main results are summarized in Table (see variance in Appendix Table 10). DeepEDM achieves state-of-the-art performance on the multivariate forecasting benchmarks, winning on 36 metrics compared to 5 for the next-best dynamical system-based method, Koopa, and 11 for the strongest deep learning model, CycleNer (Lin et al., 2024). These results highlight DeepEDMMss effectiveness and versatility across diverse domains. Notably, DeepEDM excels at the MAE metric, which is less sensitive to outliers, suggesting a stronger ability to capture underlying trends. Interestingly, the Naive baseline, outperforms all models in case of Exchange (Stocks) dataset, consistent with find- ings of (Hewage et al., 2020), thus revealing the blind spots of many forecasting models. Beyond multivariate settings, DeepEDM also exhibits strong performance in univariate forecasting on the M4 dataset (see Appendix A.3).
the ETT datasets, which contain 7 sequences, we train on sequences using only timesteps from the standard train- ing split, and test on sequences 4 6 using the standard test split. This 3 : 3 split is necessary several baseline models are unable to handle differing input dimensions between training and testing. Similarly, for the Exchange dataset (8 sequences), we train on the first 4 sequences and test on the last 4. For the Weather dataset (21 sequences), we train on sequences 0-9 and test on sequences 10-19.
Baselines. We compare DeepEDM to three representative baselines, including Koopa, iTrrnsformer, and PatchTST. The forecasting horizon H varies over (48, 96, 144, 192), with the lookback window set to 2H in all cases.
Results. Table shows the results. DeepEDM leads the performance in both MAE and MSE, ranking first in 39 out of 48 settings. The results demonstrate DeepEDM's ability to generalize across different time series.
4.3. Further Analyses
Generalization to Unseen Time Series
Robustness to Measurement Noise
Rationale. Time series forecasting benchmarks typically employ temporal splits for evaluation, that is, training on earlier time steps and testing on later ones. To evaluate the generalization across sequences, we considers a more challenging setting: splitting across different time series (i.e., channels) within the same dataset.
Rationale. We hypothesize that DeepEDM's learned latent space functions as a noise-robust kernel that more accurately preserves the local neighborhood structure of the underly- ing state space than time-delay embeddings. We conduct experiments with simulated date to verify this hypothesis.
Simulation and setup. We simulate trajectories using a chaotic Lorenz system with a = 10.0, = 28.0, 3 = 2.667, and initial conditions: (0.0, 1.0, 1.05). where the ground- truth states x, E R are known. To satisfy the univariate embedding requirement of Takens' theorem, we consider a
Setup and datasets. In addition to the standard temporal train-test split, we also partition the time series (variates) in ETT, Exchange, and Weather into disjoint training and testing sets, ensuring no overlap in sequence identity. For
8

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LETS Forecast: Learning Embedology for Time Series Forecasting
Table Model design ablation. We evaluate the effects of progressively incorporating key components into our model, with metries averaged over four prediction lengths and three random seeds (#o shown). Each successive addition yields consistent improvements across most metries relative to the preceding configuration.
Linear
MLP
MLP+EDM
Full Model
Dataset
MSE
MAE
MSE
MAE
MSE
MAE
MSE
MAE
ECL
0.1640=0:0001
0.2528++..om 0001
0.1616=0.0005
0.2632+0.000 0000004
10.000
0.2409=0.0001
0.1487=#0.00m
0.2404=0.0004
ETTh1
0.3005
0.3907 000S
0.3782=0.000n
0.3915=0.000 0001
0.3782=0.0010
0.3939=0.0002
0.3702=000033
0.3897=0.0020
ETTh2
0.2951 +0.0004
0.3403 000S
0.2910=0.0010
0.3377=0.0005
0.3017=0.0039
0.3433=0.0019
0.0054=0.0081
0.3391=0.00/
ETTml
0.31 0.0000
0.3449 OO0M
0.312320.000 0002
0.3442 0001
0.3019=00000s
0.3378 0002
0.2984=2.0004
0.3357 000000
ETTm2
0.1 0382200002
0.2602:. 1000
0.1836 0002
0.25988 0001
0.1830=00.0011
0.2585 0008
0,1817 2000015
0.2572 0008
Traffic
0,5001 00000001
0.3226 CXMT
0.4521 L.o 0016
0.3104.1..a 0014
0,3930 -00.000000022
0.2683 0015
0.4001
1:0.0004
0,2663
10063300.000 0000
Exchange
0,1097 00000
0.2247 0000
0.1110
0.2262. 0011
0.1 0.1122=00.002 00023
0.2278....om 0019
0,1090
0.2247 100.0015
ILI
2. 2,0240...000
0,9271
1:0.0163
1.9827
0.886411..mm 0271
1.6837=000oz
0.7977
0.0001
1.6779
10000091
0.7935
1.0.000000000 00887
Weather
0,0055
0.22-19=0.000
0.1899
0.2198 0000
0.1660= 0007
0 0.2002=00.007
0,1651
1:000004
0.1989
=:0. -0,0004
#lmprvvemntis
14
13
17
Baseline
#Degradations
4
Table Robustness noise. We compare time-delaedd embed- dings with our Icarned kernel for K K-nearest neighbor retrieval on simulated data, reporting mean recall as the evaluation metric.
blocks (w. MSE loss), and finally (3) the full model (w. optimized loss). This ablation allows us to isolate and quan- tify the impact of each component on overall forecasting performance. All ablation experiments are conducted on 9 standard multivariate time series benchmark datasets, using 4 different prediction lengths per dataset. Each setting is repeated with 3 random seeds. We report MSE and MAE, averaged across both prediction lengths and seeds, to ensure statistically robust and comprehensive evaluation. Addi- tional ablations on the effects of the Ltd loss, as well as the choice of lookback length, time delay, and embedding dimensions, can be found in Appendix A.5.
Recall (cleun)
Recall (noisy)
6y
Time-delayed Learned (ours)
Time- delayde
Leamed (ours)
1
0.707
0.990
0.082
0.849
K=1
5
0.986
0,990
0,257
0.957
10
0.998
0,990
0.396
0.973
7
0.545
0.586
0.220
0.527
K=7
14
0.728
0.753
0.368
0.622
28
0.896
0.857
0.564
0.730
measurement function that takes a single dimension of x as yt- All experiments are run independently for each of the 3 dimensions, and the averaged metrics are reported. For each time step i, we identify K nearest neighbors in the state space using Euclidean distance, and treat them as the ground-truth. We then retrieve top K neighbors using: (i) time-delay embeddings of ye. or (ii) via distances computed with the learned kernel in Eq. 6. We compare the retrieved neighbors against the ground-truth, and report mean recall. This is done across (EE [1,7] 7J) and similar to Section 4.1 under two noise settings: (i) noise-free, and (ii) with additive Gaussian noise (onois = 2.5).
Results. Table 3 summarizes the main ablation results (full results in Appendix Table 12 and Table 9). The simple lin- ear baseline performs the worst, confirming the inadequacy of linear models for nonlinear temporal dynamics. Intro- ducing an MLP leads to a moderate improvement, while the inclusion of EDM blocks yields significant gains across most fetausets-ddmmostraattii their effectiveness in captur- ing nonlinear and multiscale interactions. Incorporating optimized loss further refines performance, indicating the benefit of aligning the optimization objective with dynam- ical structure. Our results provide clear empirical support for each design choice in DeepEDM.
As shown in Table 4, both methods achieve high recall under noise-free conditions. However, when noise is introduced, recall for the time-delay embedding drops sharply. In contrast, our learned kernel degrades more grace- fully, maintaining significantly higher recall. This suggests that the latent space preserves the topological structure of the true state space more effectively in the presence of noise. Our DeepEDM thus offers robustness to input noise, provid- ing a key advantage over EDM real-world applications.
Results.
5. Conclusion
In this paper, we presented DeepEDM, a novel framework that integrates dynamical systems modeling and deep neu- ral networks for time series forecasting. By leveraging time-delayed embeddings and kernel regression in a latent space, DeepEDM effectively captures underlying dynamics with noisy input, delivering state-of-the-arr performance across synthetic and real-world benchmarks. Future work should explore the more advanced S-map (Chang al., 2017) method within EDM, for even greater flexibility in modeling nonlinear dynamics.
Model Design Ablation
Rationale. We conduct an ablation study to evaluate the contribution of each component in DeepEDM.
Setup and datasets. We begin with a minimal baseline consisting of a single linear layer, and incrementally add: (1) multilayer perceptron (MLP) (w. MSE loss), (2) EDM
9

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LETS Forecast: Learning Embedology for Time Series Forecasting
Acknowledgment
Dixon, P. A., Milicich, M. J., and Sugihara, G. Episodic fluctuations in larval supply. Science (New York, N.Y.), 283(5407):1528-1550 March 1999. ISSN 1095-9203. doi: 222.://ciiecce23.5500071
This work was partially supported by National Science Foun- dation under Grant No. CNS 2333491, and by the Army Research Lab under contract number INF-222220
Farmer, D. and Sidorowich, J. J. Predicting chaotic time series. Physical review letters, 59(8):845, 1987.
Impact Statement
This work presents a novel approach to time series fore- casting. The potential broader impact includes improved forecasting accuracy various domains such as economics, energy, transportation, and meteorology, leading better decision-making and resource allocation. However, it is im- portant to acknowledge that improved forecasting accuracy may also lead to unintended consequences, such as over- reliance on predictions or misuse of predictive models. It is crucial to use forecasting tools responsibly and ethically, considering potential biases in data and models, and ensur- ing transparency and accountability in their applications.
Hewage, P., Behera, A., Trovati, M., Pereira, E.,, Ghahre- mani, M., Palmieri, F., and Liu, Y. Temporal convo- lutional neural (ten) network for an effective weather forecasting using time-series data from the local weather station. Soft Computing, 24: 24:16453-16482, 2020.
Hewamalage, H., Ackermann, K., and Bergmeir, C. Forecast evaluation for data scientists: common pitfalls and best practices. Data Mining and Knowledge Discovery, 37(2): 788 832,832,100 2023.
Hochreiter, S. and Schmidhuber, J. Long short-term memory. Neural Comput., 1997.
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A. Appendix
This appendix provides additional details on several aspects of our study. First, we provide more details on the terminology essential for the background on our method (A.1). Next, we outline the details of the implementation, optimization, and training of DeepEDM (A.2). Further, we describe experiments conducted on the short-term forecasting M4 benchmark (A.3) and results from the standard lookback searching setting for long-term forecasting (A.4). Additionally, we include detailed results of experiments studying the impact of lookback length (A.5.1), sensitivity to time delay and embedding dimension (A.5.2), loss function (A.5.3) and stability of our results (A.6). Finally, we elaborate on the synthetic data experiments (A.7).
A.1. Terminology and Definitions
Definition A.1 (Manifold). A manifold M of dimension d is a topological space that is locally homeomorphic to R which means that every point in M has a neighborhood that resembles an open subset of IR If M has smooth structure, allowing for differentiation, it is called a smooth manifold.
Definition A.2 (Smooth Map). A function f: -M- - N between smooth manifolds is called smooth it has continuous derivatives all orders in local coordinates.
Definition A.3 (Homeomorphism), A function f X > Y between topological spaces is homeomorphism if it is a continuous bijection with a continuous inverse. This ensures that X and Y have the same topological structure. Definition A.4 (Diffeomorphism). A diffeomorphism is smooth function f M - N between smooth manifolds that is bijective and has a smooth inverse. If such map exists, M and N are said to be diffeomorphic, meaning they have the same smooth structure.
Definition A.5 (Immersion). A smooth map f M + N is an immersion if its differential dfp: : T - Typp) N is injective at every point E M. If f is also injective as a function, it is called an injective immersion.
Definition A.6 (Submanifold). subset S C R a submanifold it a manifold itself and the inclusion map i S - Rm an embedding. This means that S locally resembles a lower-dimensional Euclidean space and inherits a smooth structure from T
Definition A.7 (Embedding). An embedding of a smooth manifold M into RT is smooth injective immersion that is also a homeomorphism onto its image. This means that the map preserves both the local differential structure and the topology of M, ensuring that M is faithfully represented in IR" without self-intersections or distortions.
Definition A.8 (Generic Choice). A property is said to hold for a generic choice of a parameter (such as the delay t or observation function h) it holds for all choices in residual subset of the parameter space. Residual sets are dense the appropriate function space and contain a countable intersection of open dense sets, meaning that "almost every" choice satisfies the property in a topological sense.
A.2. Implementation Details
Implementation: Similar to most of the baselines, we developed DeepEDM within the popular Time-Series-Library benchmarking repository (Wu et al., 2023; Wang al., 2024b) to ensure methodological consistency with baseline approaches in data preprocessing, splitting, and evaluation metrics (MSE and MAE). We also ensure our implementation is free of the "drop last" bug as reported by (Qiu et al., 2024) that can artificially inflate evaluation metrics.
In our implementation, the base predictor f is instantiated as an MLP with I to 3 layers, each followed by non-linear activation and dropout. The number of DeepEDM blocks is also varied between and 3 based on dataset size, with larger datasets benefiting from increased expressivity through deeper architectures.
Normalization:
Following prior work (Li et al., 2023; Liu et al., 2024b; Nie et al., 2023), we also apply reversible instance normalization (Kim et al., 2022) to the input history and output predictions.
Loss function: The primary optimization objective for the DeepEDD model is minimizing the emror between the predicted forecast and true forecast mathematically formalized as:
1
Lon
pred
(9)
where y: signifies the actual value and goo represents the value predicted by the model at timestep i. For the long-term
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LETS Forecast: Learning Embedology for Time Series Forecasting
forecasting tasks, follow (Xiong al., 2024) in optimizing the first temporal difference errors (Lu) defined as:
1 H H
Ld
Σ (V Vitti
(10)
pood Here, Vyt-i and Vutti denote the true and predicted first differences (i.e. Vt+1 y1 and Vi+1 v/= )), respectively. The function evaluates the mean absolute error for these differences, thus focusing on the accuracy of sequential changes of the series. Further following the methodology proposed by (Xiong et al., 2024), we also consider the balance between these loss components using X, defined as:
pred
H Σ (ssnn((
(Vtt+)) + sgn( (gn()))))))
(11)
Here sgn refers to the Signum function. The final composite loss function L is then computed as weighted sum of Ler and Cu, modulated by A:
C= Ler + (1 A) Laa
(12)
The parameter À dynamically adjusts the weighting between the fer and Lut based on the frequency of sign changes between the actual and predicted differences, promoting higher fidelity in capturing dynamic temporal patterns. For more details on Lu loss, we refer the readers to (Xiong et al., 2024).
In our experiments, we set Len to Mean Absolute Error (MAE) for the benchmarking tasks. We hypothesize that MAE is more suitable because DeepEDM aims to model the underlying dynamics and thus focusing on the general trend aligns better with this objective, avoiding excessive sensitivity noisy outliers. However, for ECL and Traffic datasets which have high dimensionality and are generally noisier, MAE does not perform competitively, For these specific cases, we instead set Ler to Mean Squared Error (MSE) loss.
Training: DeepEDM is trained for 250 epochs using the AdamW (Loshchilov & Hutter, 2017) optimizer with learning rate of 0.0005 and a batch size of 32. Following standard practices in time-series forecasting, an early stopping mechanism based on validation set performance metrics is implemented to mitigate overfitting.
A.3. Short-term Forecasting Experiments and Results
We now present the comprehensive evaluation results of the DeepEDM on the popular short-term univariate forecasting M4 benchmark. This benchmark consists of six datasets, each corresponding to a different frequency: yearly, quarterly, monthly, weekly, daily, and hourly. For our experiments, we follow the standard setup of all the reported baselines, where the lookback length is set to twice the forecast length H e [6, 48). Consistent with prior works, DeepEDM optimized using the SMAPE loss function.
Following standard baselines, we use the Symmetric Mean Absolute Percentage Error (SMAPE), MAPE (Mean Absolute Percentage Error), Mean Absolute Scaled Error (MASE), and overall weighted average (OWA) metrics to evaluate the forecasting performance. For brevity, we only provide the formulation of these metrics and refer the reader to (Oreshkin et al., 2020) for more details:
Metrics:
200 H
SMAPE
Σ
0T++
Dr++l
MAPE H
100
Σ
YT+: Or+il
H
MT++ + 10++1
VT+:l
1=1
1
BT Dr+al
1
SMAPE
MASE
MASE
OWA
H 1=1
2j=++1 |/
2
SMAPExaiveR
MASE
Results:
The results of our experiments, summarized Table 5, demonstrate that DeepEDM outperforms the dynamical modeling-basee methods on all subsets. also surpasses all methods within the three subsets grouped under the "others"
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LETS Forecast: Learning Embedology for Time Series Forecasting
category, while delivering competitive performance across other subsets. The notable success in the "others" category can be attributed to the typically longer sequences found in these subsets, which better facilitate the reconstruction of the underlying dynamical system. In contrast, other subsets contain shorter sequences, which pose challenges to effective system reconstruction. Nonetheless, DeepEDM again exhibits competitive performance (best Weighted Average SMAPE for all datasets) across this benchmark, further showcasing its capabilities.
Table 5. Univariate forccasting results on M4 dataset. The M4 dataset comprises six datasets, three of which are included in the "Others" category. These three subsets generally contain longer sequences, allowing our method to perform better and achieve superior performance compared to all other methods these subsets. All prediction lengths are in [6, 48). Baseline results are from Koopa (2024b) and TimeMixer (2024a). Bold represents the best values while underline represents 2nd best. Gray represents dynamical modeling based
methods.
Models
Dec
Knopa
KNF
TimeMixer NemesNet N.HSSS
N.BEATS'
PatchTST
FiLM
LighiTS
DLincar
FED.
Stationary
Auto.
Pyra.
Yeady
SMAPE
13.243
13.352
13,986
13,206
13.387
13.418
13,436
16.463
17.431
14.247
16,965
13.728
13.717
13:974
15,530
14.727
MASE
2,973
2.997
3.029
2.916
2.996
3.045
3.043
3,967
4.043
3.109
4.283
3,048
3.078
3.134
3.711 3.418
OWA
0.779
0.786
0.804
0.776
0.786
0.793
0.794
1.003
1.042
0.827
1.058
0.803
0.807
0.822
0.942
0.881
Quaneetyy
SMAPE
10.04
10.159
10.343
9.996
10.100
10.202
10.124
10.644
12.925
11.364
12.145
10.792
10,958
11.338
15.449
11.360
MASE
1.177
1.189
1.202
1.166
1.182
1.194
1.669
1.278
1,664
1.328
1.520
1.283
1.325
1,365
2.350
1.401
OWA
0.885
0.895
0.965
0.825
0.890
0.899
0.886
0.949
1.193
1.000
1.106
0.958
0.981
1.012
1.558
1.027
Mombly
SMAPE
12.547
12.730
12.894
12.605
12.670
12.791
12.677
13,399
15.407
14,014
13,514
14,260
13,917
13.958
17.642
14.062
MASE
0.933
0.953
1.023
0,919
0.933
0.969
0.937
1.031
1.298
1.053
1.037
1.102
1.097
1,103
1.913
1.141
OWA
0.873
0.901
0.985
0.869
0.878
0.899
0.880
0.949
1.144
0.981
0.956
1.012
0.998
1.002
1.511
1.024
Others
SMAPE
4.339
4.861
4.753
4.564
4.891
5.061
4,925
6,558
7.134
15,880
6.709
4.954
6.302
5,485
24.786
24.460
MASE
3.042
3.124
3.138
3.115
3.302
3.216
3.391
4,511
5.09
11.434
4.953
3,264
4.064
3.865
18.581
20.960
OWA
0.936
1.004
1.019
0.982
1,035
1,040
1,053
1,401
1,553
3,474
1,487
1,036
1.304
1.187
5,538
5,879
Weighted Average
SMAPE
11.695
11.863
126
11.723
11.829
11.927
11.851
13.152
14,863
13.525
13.639
12,840
12,780
12,909
16.987
14.086
MASE
1.566
1.595
1.641
1.559
1,585
1.613
1.559
1.945
2.207
2.111
2.095
1.701
1.756
1.771
3.265
2,718
OWA
0.841
0.858
0.874
0,840
0.851
0.861
0.855
0.998
1.125
1.051
1.051
0.918
0.930
0.939
1,480
1.230
The original paper of N-BEATS (2020) udopts a special ensemble method to promote the performance. For fair comparison, authors of Mixer (2024a) removed the ensemble and only compared the pure forecasting models.
A.4. Long-term Forecasting with Lookback Search
In this section, we present the forecasting results under the lookback search setting, commonly adopted in recent works such as TimeMixer (Wang et al., 2024a). This setting allows models to select an optimal lookback length from predefined set, ensuring a fair comparison while potentially benefiting methods that can leverage longer historical dependencies. In this setting, each model is evaluated on four prediction horizons (H E [96, 192, 336, 720)). with the cest-performing lookback chosen from [96, 192, 336, 512].
While this setup provides flexibility, it can be particularly challenging for dynamical systems-based methods like DeepEDM, which rely on sufficiently long lookbacks reconstruct the underlying attractor accurately. Some configurations require forecasting 720 steps into the future using only 512 steps of history, a scenario that may not always capture the full state-space dynamics. Nonetheless, as shown in Table 6, DeepEDM demonstrates strong performance, achieving 45 wins compared to 34 for the second-best model, further highlighting its robustness even under challenging settings.
A.5. Ablation Studies
In addition to the component-we component-wise ablation presented in the main paper, we conduct further experiments to evaluate additional design choices underlying DeepBDM.
1. ABLATION STUDY ON LOOKBACK LENGTH
In this section, we present an ablation study to investigate the impact of varying the lookback length on forecasting performance, evaluated across three datasets: ETTh1, ETTm2, and Exchange. The results, shown in Figure 4 plot the Mean Squared Error (MSE) against input sequence lengths, with the prediction horizon fixed at H = 96. Across the datasets, we observe that increasing the lookback window improves forecasting accuracy up to threshold, typically at T = 512. Beyond this point, performance degrades, with further increases in lookback length resulting in higher errors. This decline is attributed to a distribution shift, where the model starts to capture data points from the past that no longer align with the distribution of more recent data, introducing irrelevant or outdated information that negatively impacts forecast quality.
Notably, our model consistently outperforms the compared benchmarks across varying input sequence lengths, demonstrating
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LETS Forecast: Learning Embedology for Time Series Forecasting
Table 6. Multivariate forecasting results under the lookback search setting. Bold indicates the best performance, while underline indicates
the 2nd best. Baseline results are taken from (Wang et al., 2024a) while Naive was reproduced by us.
Ours
TimeMixer
PaccTTT
TimesNet
Crossformer
MICN
FiLM
DLineur
FEDfomeer
Stationary
Autoformer MSE MAE
Naive
Models
MSE MAE|
MSE MAE|
MSE MAE
MSE MAB
MSE MAE
MSE MAE
MSE
MAE
MSE MAE MSB MAE MSE MAE
MSE
MAE
0.145
0.183
0.147
0.197
0.149 0.198
0.172 0.220
0.232 0.302
0.161 0.229
0.199 0.262
0.176
0.237
0.217 0.296
0.173 0.223
0.266 0.336
0.259 0.254
Weather
192
0.189
0.226
0.189
0.239
0.194
0.241
0.219 0.261
0.371 0.410
0.220
0.281
0.228 0.288
0.220
0.282
0.276 0.336
0.245 0.285
0.307 0.367
0.309 0.292
336
0.240
0.267
0.241
0.280
0,306
0,282
0.246
0.337
0:495
0.515
0.278
0,351
0.267 0.323
0.265
0.319
0.339
0.380
0.321
0.338
0.359
0.395
0,376 0,338
720
0.314
0.322
0.310
0.330
0.314
0,334
0.365
0,359
0.526
0.542
0.311 0.356
0.319 0.361
0.323
0.362 0.403
0.428
0.414
0.410
0.419
0.428
0,465 0.394
Avg
0.222
0.249
0.222
0.262
0.241
0.264
0.251 0.294
00,466
0.442
0.242
0.299
0.253 0,309
0.246
0.300 00.00 0.360
0.288
0.314
0.338
0.382
0.352 0.319
96
0.178
0.199
0,167
0.220
0.224
0.278
0.219 0.314
0.181
0.240
0,188 0.252
0.320 0.339
0.289
0.377 00.201 0.304
0.321 0.380
0.456
0:446
1.539 0.816
Solar- Energy
192
0.191
0.209
0.187
0.249
0,253
0,298
0.251 0.322
0.196
0,252
0.215
0,280
0.360
0,362
0.319
0.397 0.2237 0.337 0,346 0.369)
0,588
0.561
1,360 0.735
336
0.206
0.216
0.200
0,258
0.273
0.306
0.246 0.337
0.216
0.243
0.222 0.267
0.398 0,375
0.352
0,415 0.2254 0.362
0.357 0.387
0.595 0.588
1,430 0.766
720
0.254
0.245
0.215
0.250
0:272
0,308
0:280
0,363
0.220
0.256
0:226
0.264
0.399
0.368
0.356
0:412 0.280 0.397 0.335 0.384
0.733 0.633
1,474 0,784
Avg
01.007
0.217
0.192
0.244
0.256
0.298
0.244 0.334
0.204
0.248
0.213 0,266
0.369
0,361
0.329
0,400 00.243 0.350
0.:440 0.380
0.593 0.587
1,451 0.775
96
0.133
0.228
0.129
0.224
0.129 0.222
0.168 0.272
00.150
0,251
0.164 0.269
0.154 0.267
0.140
0.237 00.193 0.308 00:169 0.273
0.201 0.317
1,588 0.946
Elecmieity
192
0.151
0.246
0.140
0.220
0.147 0,240
0,184 0,322
0.161
0,260
0.177 0,285
0.164 0.258
0.153 0.249 0:201 0.315 0.182 0.286
0.222 0.334
1,596 0.951
336
0.167
0.262
0.161 0.255
0.163 0.259
0.198 0,300
0.182
0.281
0.193 0,304
0.188 0.283
0.169
0.267 0.214 0.329 0.200 0.304
0.231 0.338
1.618 0.961
720
0,205
0.293
0.194
0.287
0.197
0.290
0.220 0.320
0.251 0.339
0.212 0.321
0.236
0.332
0.203
0.301 0.246 0.355
0.222 0.321
0.254 0.361
1.647 0.975
Avg
0.1664
0.257
0,156 0.246
0.1.5 0.253
0.192 0.295
(0.166 0,283
0.186 0.295
0.186 0.285
(6.166
0.264 00.214 0.321 (0.213 0.296
0.227 0.338
1.612 0,958
96
0.360
0.252
0.360
0.249
0.360
0.249
0.593 0,321
0.514
0.267
0.519
0.309
0,416
0.294
0.410
0.282 00.587 0.366 0.612 0.338
0.613 0,388
2,715 1.077
Traffic
192
0.375
0.255
0.375
0.250
0,379
0.256
0.617 0.336
0.549
0,252
0.537
0.315
0.408
0.288
0:423
0.287 0.604 0.373 0.613 0.340
0.616 0.382
2.747 1.085
336
0.410
0,280
0.385
0.270
0.392
0.264
0.629 0.336
0.530
0,300
0.534
0.313
0.425 0.298
0.436
0.296 0.621 0.383 0.618 0.328 0.622 0.337
2.788 1.094
720
0.458
0.31
0.430
0.281
0.432
0.286
0.640 0.350
(.573
0.313
0.577
0.325
0.520
0.353
0.466
0.315 0.626 0.382 0.663 0.355 0.660 0.408
2,810 1.097
Avg
0,401
0.275
0.387
0.262
0.391
0,264
0,620
0,336
0.542
0,283
0.541
0,315
0.442 0,308
0.434
0.295 0.609
0.376 0.6624 0. 0.340
0.628 0.379
2,765 1.088
0.356
0.384
0.361
0.390
0,370
0,400
0.384
0.402
0.418
0.438
0.421
0.431
0.422
0.432
0.375
0.399
0.376
0.419 0.513 0.491 0.449
0.459
1,294 0.713
192
0.398
0.417
0.409
0.413
0.429
0.436
0.429
0.539
0.517
0.474
0.487
0.462 0.458
0.405
0.416 0.420
0.448 0.534 0.504 0.500 0.482
1.325 0.733
336
0.419
0.425
0.430
0.429
0.422
0,440
0.638
0.469
0.709
0,638
0.569
0.551
0.501
0.483
0.439
0.443 0.459
0.465
0.588
0.535
0.52. 0.496
1.330 0,746
720
0.434
0.451
0.445
0.460
0.447
0.468
0.521
0,500
0.733
0.736
0.770
0.672
0.544
0.526
0.472
0.490
0.506
0.507
00.643
0.616
0.514
0,512
1,335 0.755
Avg
0.402
0.419
0.411
0.423
0.413
0.434
0,458
0,450
0.600
0.557
0.558 0.535
0.482
0.475
0.423
0.437
00.40
0.460
0.570
0.536
0.496 0.487
1,321 0.737
96
0.275
0.332
0.271
0.330
0.274
0.337
0.340
0.374
0:425
0.463
0:299
0.364
0.323
0.370
0.289
0.353 0.346
0.388
0:476 0:458
0.358
0.397
0.432 0.422
E17h2
192
0.341
0.374
0.317
0.402
0.314
0,382
0.231
0.322
0.473
0.500
0.441
0.454
0.391
0.415
0.383
0.418 0.449 0.439
0.512 0.493
0.456
0.452
0,534 0,472
336
0,360
0.393
0.332
0.396
0.329
0.384
0:452
0,452
0.581
0.562
0.654
0,567
0.415
0,440
0.448
0.465
0.496
0.487
0.552
0.551
0.482 0.486
0.597 0.511
720
0,386
0.424
0.342
0.408
0.379
0.422
0.462
0.468
0.775
0.665
0.956
0.716
0.441
0.459
0.605
0.551 0..463
0.474
0.562
0.560
0.515 0.511
0.595 0.519
Avg
0,341
0.381
0.316
0.384
0.371 0.404
(0.564
0,548
0.588
0.525
0.393
0.421
0.433
0.447
0..43
0.447
0.526
0.516
0.453
0.462
0.539 0.481
96
0.289
0.331
0
0.340
0.293
0.346
0.338 0,375
0.361
0.403
0,316
0.362
0.302 0.345
0.299
0.343
0..79
0.419
0.386
0.398
0.505 0.475
1,214 0.665
Er1m1
192
0.321
0.351
0.327
0.365
0.333
0,370
0.374 0.387
p.387
0.422
0.363
0.390
0.338
0.368
0.335
0.365
00.426 0.441
00.459
0.444
0.553 0.496
1,261 0.690
336
0.361
0.377
0.360
0.381
0.369
0.392
0.410
0.411
0.605
0.572
0.408
0.426
0.373
0.388
0.369
0.386
0..445
0.459
0.495
0.464
0.621 0.537
1,287 0.707
720
0.414
0.406
0.415
0.417
0.416
0.420
0.478
0.450
0.770
0.645
0.481
0.476
0.420
0.420
0.425
0.421 0..543
0.490
0.585
0.516
0.671 0.561
1.322 0.730
Avg
0.346
0.366
0.375 0.348
0.353
0.382
0.353 0.382
0.514
0.510
0.392
0:413
0.358
0.380
0.357
0.379 0.448
0.452
00:481
0.456
0.588
0.517
1.271 0.698
96
0.164
0.245
0.164
0.254
0.166
0,256
0.187 0.267
0.275
0.358
0.179
0.275
0.165
0.256
0.167
0.260 0.203 0.287
0.192
0.274
0.255 0.339
0,267 0.328
192
0.221
0.287
0.223
0.295
0.223
0.296
0.249 0.309
0.345
0.400
0.307
0.376
0.222
0.296
0.224
0.303
0.269
0.328
0.280 0.339
0.281 0.340
0.340 0.371
336
0.270
0.321
0.279
0.330
0.329
0.321 0.351
0.657
0.528
0.325
0.388
0.277
0.333
0.281
0.342
0.325
0.366
0.334
0.361
0.339
0.372
0.412 0.410
720
0.347
0.371
0.359
0.383
0.362
0,385
0,408
0.403
1.208
0.753
0.502
0.490
0.371
0.389
0.397
0.421
0.441
0.415
0.417
0.413
0.422
0.419
0.522 0.466
Avg
0.251
0.306
0.256
0.315
0.256
0,317
0.291
0.333
0.621
0.510
0.328
0.382
0.259
0.319
0.267
0.332
00.304
0.349
0.3006
0.347
0.324
0.368
0,385 0,393
Count
45
34
2
0
0
0
0
16

---

LETS Forecast: Learning Embedology for Time Series Forecasting
E11h1
ETTm2
Exchange
The Nleer
0.150
Duns
0.110
0.100
0.180
5.100
2.035
2.035
Esput Sequence Length
Inguz Sequence Length
lagut Sequence Longth
Figure 4. Impact of lookback length forecast accuracy. The prediction horizon is fixed at H 96, while the input sequence length T' E {192, 288, 336, 512, 720) is varied to assess its effect on forecasting performance. Increasing the lookback window generally improves accuracy up certain point; however, excessively long lookbacks can introduce irrelevant information noise, ultimately degrading performance.
its robustness in different temporal contexts. This study highlights the importance of a balanced lookback window in optimizing forecasting models. While short windows may lack sufficient context, excessively long windows risk overfitting to irrelevant historical trends. Our results suggest that moderate lookback window length (e.g., T 512) offers the best trade-off between context and relevance, evidenced by the performance drop beyond this threshold.
A.5.2. SENSITIVITY TO TIME DELAY AND EMBEDDING DIMENSION
We conducted additional experiments to investigate the sensitivity of DeepEDM to two key hyperparameters that govern the time-delay embedding: the embedding dimension 67 and the delay interval т. Together, these parameters define how historical observations are mapped into the delay-coordinate space. Specifically, a time-delay embedding with dr 3 and results in x; [Tr, 2t-2, Tz-4].
1 yields a 3-dimensional vector x; [Te, -1 21-2], while 8T and
In our main experiments, is fixed to 1, and of is selected empirically per dataset. To assess the robustness of the model, we perform ablation studies by varying one parameter while keeping the other fixed. For the dy-sensitivity experiments, we fix and vary d7 E 32,3, 4, 5). For the r-sensitivity experiments, we fix ôT and explore T E {1, 2, 3, 4].
Effect of Embedding Dimension 87. The impact of the embedding dimension 8f is summarized Table 7. The results show that its influence is highly dataset-dependent. In several cases, performance remains relatively stable across different values of ôr, indicating that the underlying system may be either intrinsically low-dimensional or already adequately represented by the chosen embedding. When the dynamics lie low-dimensional manifold, larger values of 57 become redundant. Conversely, in high-ding -itheiinnniinall systems, small ōT may lead to underembedding, resulting in similar but suboptimal performance across configurations. These observations align with classical results from delay-cooe embedding theory.
Effect of Delay Interval r. Table 8 presents the results for varying the delay interval т. We observe that r = 1 consistently yields the best or near-best performance across all datasets. This also the case considered Takens' theorem. Nevertheless, determining the optimal pair (δr, r) remains open problem. Developing principled strategies for joint selection may further improve model accuracy, which we leave for future work.
A.5.3. ABLATION STUDY ON THE LOSS FUNCTION
To study the role of the loss function, we compare the full DeepEDM model trained with our full loss function to a variant trained solely with standard MSE. The results this experiment, detailed in Table 9, reveal that while incoporating Lia loss generally leads to lower errors. MSE occasionally performs comparably or even slightly better suggesting complementary strengths. Importantly, the full model consistently delivers the best overall performance.
17

---

LETS Forecast: Learning Embedology for Time Series Forecasting
Table 7. Ahlation i.e. embedding size
Dalaset
by
b7
by T
67
11
67 15
MSE
MAI
MSI
MAE
MSE
MAE
MSE
MAE
MSI:
MAE
48
0.3288
0.3587
0.3224
0.3561
0.3245
0.3576
0.3240
0,3582
0.3236
0.3583
BTTb
96
0.3656
0.3835
0.3708
0.3865
0.3688
0.3843
0.3686
0.3858
0.3715
0.3866
144
0,3922
0.3993
0.3881
0.3978
0.3886
0.3980
0.3915
0.3995
0.3918
0,4000
192
0.4064
0.4152
0.4046
0.4147
0.4017
0.4132
0.4258
0.4327
0.4593
0.4526
48
0.2263
0.2886
0.2276
0.2902
0.2238
0.2871
0.2231
0.2872
0.2236
0.2869
Er
96
0.2917
0,3341
0.2870
0.3320
0.2881
0.3328
0.2905
0,3335
0.2940
0.3344
0.3307
0,3635
0,3225
0.3603
0.3224
0.3611
0.3270
0,3631
0.3352
0.3666
192
0.3523
0.3783
0,3516
0.3764
0.3450
0.3741
0.3505
0.3763
0.3451
0.3765
48
0.2822
0.3223
0.2827
0.3224
0.2775
0.3191
0.2782
0.3175
0.2784
0.3182
ET/ml
96
0.2902
0.3287
0.2874
0.3273
0.2878
0,3277
0,2904
0,3290
0.2904
0.3280
144
0.3104
0.3441
0.3084
0.3445
0.3041
0.3428
0.3087
0.3439
0.3034
0.3422
0.3226
0.3533
0.3200
0.3523
0.3213
0.3524
0.3257
0.3545
0.3226
0.3537
48
0.1344
0.2219
0.1334
0.2213
0.1335
0.2211
0.1332
0.2212
0.1337
0.2211
ET1m2
96
0.1689
0.2473
0.1692
0.2478
0.1697
0.2483
0.1691
0.2478
0.1670
0.2470
144
0.1991
0.2707
0.2048
0.2727
0.1974
0.2685
0,2023
0.2722
0.2057
0,2727
192
0.2272
0,2886
0.2244
0,2890
0.2248
0.2881
0.2288
0.2904
0.2291
0.2911
48
0.0429
0.1429
0.0418
0.1404
0.0429
0.1414
0.0443
0.1458
0.0430
0.1433
Exchange
96
0.0894
0.2087
0.0854
0.2026
0.0825
0.2008
0.0861
0.2039
0.0886
0.2061
144
0.1348
0,2586
0.1298
0.2529
0.1326
0.2553
0.1427
0.2664
0.1323
0.2560
192
0.1801
0.3035
0.1931
0,3144
0.1785
0.2996
0.1854
0.3085
0.1777
0,3000
48
0.1404
0.1695
0.1421
0.1735
0.1371
0.1668
0.1396
0.1686
0.1424
0.1753
Weather
96
0.1600
0.1948
0.1573
0.1916
0.1579
0.1927
0.1578
0.1920
0.1578
0.1919
144
0.1741
0.2099
0.1735
0.2092
0.1741
0.2096
0.1749
0.2098
0.1744
0.2094
0.1910
0.2262
0.1910
0.2263
0.1911
0.2263
0.1911
0.2263
0.1911
0.2264
Table 8, Ablation on
Dataset
==1
==2
T=
MSE
MAE
MSE
MAE
MSE
MAB
0.3246
0.3575
0.3270
0.3589
0.3270
0,3602
F7TT
96
0.3689
0,3844
0.3663
0.3839
0.3653
0,3836
144
0.3885
0.3980
0.3902
0.3988
0.3911
0.3985
192
0.4017
0.4129
0.4038
0.4164
0.4090
0.4194
48
0.2238
0.2871
0.2269
0,2879
0.2240
0,2870
1:71h2
96
0.2882
0.3333
0.2885
0.3331
0.2875
0.3326
144
0.3224
0.3611
0.3290
0.3652
0.3316
0.3653
192
0.3450
0.3741
0.3541
0.3787
0.3583
0.3803
48
0.2782
0.3195
0.2795
0,3189
0.2804
0.3198
k11ml
96
0.2869
0.3282
0.2901
0.3288
0.2866
0.3279
144
0.3053
0.3439
0.3094
0.3457
0.3079
0.3452
192
0.3213
0.3523
0,3209
0.3527
0.3203
0.3530
48
0.1337
0.2212
0.1342
0.2222
0.1346
0.2230
FT1m2
0.1700
0.2485
0.1704
0.2485
0.1706
0.2479
144
0.1965
0.2684
0.1967
0,2690
0.1952
0.2681
192
0.2220
0.2871
0.2246
0.2881
0.2231
0.2873
0.0429
0.1414
0.0426
0.1412
0.0423
0.1410
Exchunge
96
0.0829
0.2011
0.0835
0.2020
0.0838
0.2023
144
0.1340
0.2567
0.1336
0.2564
0.1317
0,2547
192
0.1769
0.2991
0.1773
0.2992
0.1789
0.2999
48
0.1371
0.1668
0.1397
0.1690
0.1406
0.1695
Weather
96
0.1581
0.1926
0.1580
0.1931
0.1587
0.1938
144
0.1741
0.2096
0.1740
0.2096
0.1749
0.21
192
0.1938
0.2280
0.1969
0.2310
0.56
0.2308
18

---

LETS Forecast: Learning Embedology for Time Series Forecasting
Table Ablation study the loss function.
Dataset
De
DeepEDM (with MSE loss)
MSB
MAE
MSE
MAE
48
0.1610
0.2470
0.1591
0.2463
96
0.1370
0.2320
0.1380
0,2322
144
0.1450
0.2390
0.1466
0.2393
192
0.1510
0.2440
0.1528
0,2458
48
0.3240
0.3570
0.3322
0.3738
ETThi
0.3650
0.3840
0.3697
0.3982
144
0,3880
0,3980
0.3990
0,4095
192
0.4070
0,4210
0.4061
0.4205
0.0420
0.1420
0.0415
0.1403
Exchunge
96
0.0880
0.2050
0.0827
0,2025
144
0.1330
0.2550
0.1297
0.2552
192
0.1780
0.3010
0.1739
0,3000
48
0.2250
0,2880
0.2265
0,2958
ETTTZ
96
0.2890
0.3330
0.2885
0.3430
144
0.32440
0.3620
0.3242
0.3679
192
0.3510
0.3770
0.3547
0,3888
48
0.4480
0.2860
0.4374
0,2863
Traliic
96
0.3830
0.2590
0.3757
0.2608
144
0,3800
0.2580
0.3781
0,2615
192
0.3870
0.2620
0.3809
0,2644
0.2770
0.3180
0,2820
0.3319
ET1ml
96
0.2880
0.3280
0.2882
0.3425
144
0.3080
0.3440
0.3106
0.3581
192
0.3220
0.3530
0.3233
0.3646
48
0.1330
0.2210
0.1350
0,2309
ET1m2
0.1690
0,2480
0.1676
0,2548
144
0,2030
0.2710
0.1991
0.2774
192
0.2240
0.2890
0.2205
0,2924
48
0,1380
0.1680
0.1376
0.1774
Wealber
96
0.1570
0.1920
0.1551
0,1998
144
0.1740
0.2100
0.1733
0.2201
192
0.1910
0.2260
0.1909
0,2392
A.6. Stability of Results
To assess the robustness of our main results on the standard multivariate forecasting benchmark (Table 1), evaluate the stability of each metric by computing its standard deviation across five independent random seeds. The detailed results are presented in Table 10. As shown, the standard deviations are consistently low across most datasets, indicating that DeepEDM yields stable and reliable forecasts. As expected, the ILI dataset, being much smaller, exhibits relatively higher variance.
A.7. Additional Details on Synthetic Data Experiments
Data Generation: To systematically analyze model performance under deterministic yet unpredictable systems, we generate synthetic datasets for both chaotic and non-chaotic dynamical systems. Non-chaotic systems exhibit predictable behavior, where small variations in initial conditions result in only minor deviations in long-term trajectories. In contrast, chaotic systems, despite being governed by deterministic rules, exhibit extreme sensitivity to initial conditions, leading to exponentially diverging trajectories over time. canonical example of deterministic chaos, the Lorenz system, is governed by set of nonlinear differential equations that give rise to strange attractor, characterized by series of bifurcations and highly sensitive trajectory evolution (Figure 5, middle row).
To capture distinct dynamical regimes of the Lorenz system, we generate two configurations: (i) Chaotic behavior: σ = 10.0, = 28.0, 3 = 2.667 and initial conditions: (0.0, 1.0, 1.05) (ii) Non-chaotic behavior: o = 10.0, = 9, 2.667, and initial conditions: (10.0, 10.0, 10.0). For the Rössler system, we generate one chaotic configuration using 0.2, 0.2, e = 5.7 and initial conditions: (1,,1.,1) 1,,
19

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LETS Forecast: Learning Embedology for Time Series Forecasting
Table 10. Standard deviation of the main results (Table 1) over five seeds across different forecast horizons: E {48, 96, 144, 192) for all datasets, except ILI, where H E {24, 36, 48, 60). The results demonstrate stability, with consistently low standard deviations across datasets. However, the ILI dataset, being significantly smaller, naturally exhibits relatively higher variance, particularly for shorter forecast horizons.
Forecast Length (If)
Dataset
Ho
H
H2
H3
CMSE
GMAE
CASE::
OMAE
AMAA
JASE
BCL
0.0007
0.0006
0.0006
0.0004
0.0008
0.0008
0.0003
0.0005
E
0.0015
0.0009
0.0038
0.0008
0.0017
0.0006
0.0055
0.0050
ETTh2
0.0018
0.0009
0.0013
0.0012
0.0081
0.0033
0.0077
0.0036
ETTml
0.0010
0.0005
0.0006
0.0005
0.0020
0.0005
0.0014
0.0008
ETTm2
0.0004
0.0003
0.0010
0.0005
0.0028
0.0016
0.0023
0.0013
Traffic
0.0033
0.0008
0.0007
0.0009
0.0008
0.0007
0.0012
0.0012
Exchange
0.0006
0.0013
0.0021
0.0020
0.0038
0.0036
0.0041
0.0026
11.1
0.1383
0.0272
0.0642
0.0104
0.0380
0.0103
0.0621
0.0141
Weather
0.0007
0.0009
0.0003
0.0003
0.0005
0.0003
0.0005
0.0003
The initial conditions were selected to ensure trajectories remain well-defined and not excessively perturbed even under the highest noise settings. Further to systematically evaluate model performance in presence of noise, we simulate noisy conditions by introducing Gaussian noise (0,, nooiee) with noise E {0.0, 0.5, 1.0, 1.5, 2.0, 2.5) with higher Anisse denoting higher levels of noise. This results in a total of 18 synthetic datasets (3 systems x 6 noise levels). The underlying dynamical systems and noise levels are illustrated in Figure 5.
Each synthetic dataset is divided into sequential non-overlapping training, validation, and testing splits. The models are trained on their respective training sets and evaluated on the test sets, with validation sets used for early stopping to mitigate overfitting.
Experimental Setup:
All based models are trained to forecast a fixed number of future steps (96 steps) based on fixed lookback window (192 steps). The performance of each model is assessed based on its ability to forecast accurately over varying lengths (p) and under different noise conditions. Specifically, while models are trained with a forecast length of 48 steps, only the first p steps of each forecast are considered during testing. This evaluation strategy ensures a consistent and unbiased comparison across different prediction lengths and models.
Results: Table 11 details the quantitative results of our experiments on synthetic datasets under varying noise levels (anoise) and prediction horizons (H) across three dynamical systems. DeepEDM consistently delivers the lowest MSE and MAE, demonstrating superior forecast performance, particularly in noisy and chaotic regimes.
Chaotic regimes pose significant challenges for long-term forecasting due to their inherent complexity, leading to relatively high errors. Despite these challenges, DeepEDM handles forecasting far more effectively than the baseline methods. At a = 2.5 and H = 48, its MSE (17.267) is 40% lower than Koopa (28.804), 45% below iTransformer (31.599), and 60% below Simplex (43.548). This advantage is also evident in the no-noise regime (Gnoiee = 0, H 48), where DeepEDM achieves MSE of 10.467 outperforiig Koopa (18.978) by 44.85%, illansformer (18.531) by 43.52%, and Simplex (30.985) by 66.22% DeepEDM also demonstrates consistent superiority on the Rössler system, particularly in terms of MAE, further reinforcing its robustness against noise and its ability to model complex nonlinear dynamics.
Chaotic datasets:
Non-chuotic datasets: In the simpler non-chaotic setting with no noise, all the baselines perform comparably, however as the noise level increases DeepEDM still maintains a competitive edge. For instant, at a 220 and H = 48, it achieves an MSE of 0.048-18% lower than Koopa (0.059) and 6% lower than iTransformer (0.051), while Simplex deteriorates drastically to 3.921.
In summary, across all three systems, DeepElDM outperforms both classical EDM methods and learning based baselines, showcasing its resilience under noise and across varying prediction horizons.
20

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LETS Forecast: Learning Embedology for Time Series Forecasting
-10
-30
-32
Figure 5. Visualization of the synthetic datasets: Non-chaotic Lorenz (top row), chaotic Lorenz (middle row), and chaotic Rössler (bottom row) systems. As the noise level increases (from left to right), forecasting future states becomes progressively more challenging. Pink dots indicate the new attractor under the current regime, while the light blue denotes the original attractor for reference.
21

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LETS corccasts Learning Sowedooooyy for Time Series corecasting
Table 11. Multivariate forecasting results with prediction lengths H E [1, 5, 15, 48) for multiple dynamical systems. The amount of Gaussian noise added is lincarly inereased by varying the standard deviation.
MSE MAE
0.0I0 0,000 0,000
0.395
0.789
1.5 0.025 0.126 0.925 12126 0,026 1.127 0,000 0.535 0,050 O. 136 0.030 0.135 0032 0.139 0.000 153 0.027 031 0.028 1.3 0,30 0.136 0004 0.145 22088 1.185 2.208 1 2.208 1.185 2.206 1.184
1.578
1,973
3,952
4.041
4.236
1.5 0:979 0:733 1.937 0,932 5444 1.325 14,675 2.017 4.682 1.994 5A33 1835 11.317 2,343 24,270 3.297 6,140 1.99 7:232 2015 11164S 2,391 22,689 3.184 12.318 2.635 16.095 2,999 55.392 3,712 36.597 4,455
4.711
4,982
0.9 0.028 0.067 0,030 0,069 0,033 0.074 0,291 0.132 0.088 0.127 0,042 0.121 0.035 0.114 0.299 O. 0.170 0,089 0.178 0.085 0.171 0,107 0.171 0.684 0.225 1.029 0.647 1.096 0.668 1.350 0.724 2,388 0.897
1.477 0,889 1,528 D. 0.906 1.811 0,959 3,000 1.137
4313 1.475
4.73 1.625 4,354 L68 4.694 1.682 6.191 1.846
2.079 8.605 2.233
2.625
2.55422.255
48
0.5 0,004 0.049 0,004 0,050 0,004 0,050 0.005 0,033 0.0044 052 0,004 0,052 0.005 0,054 0.007 0.0G1 0.005 0,054 0.005 0.057 0.005 0.058 0.006 0.062 0.245 0. 0.395 0,245 O. 0.395 0.245 0,395 0.245
1.0 0.013 0.091 0,014 0.092 0,014 00,093 0.017 0.099 0.014 0.092 0.014 0.092 0,15 0,035 0.019 0.106 0015 0.098 0,016 0.100 0016 0.102 0019 1. 0981 0,790 0.981 0,790 0.981 0,790 0,980
1.580 3,926 1.580 1,921
1.944 6.134 1.974 6.127
0.0 0.056 0.130 0.3 3300 0L320 24144 0,763 10.476 1613 1.772 0.912 2.276 09966 6.132 1.444 18.978 2600 3.044 1,229 3.995 1,344 ROIS 1.753 18.531 2.625 R696 2,048 12,259 2.458 20 885 3.221 30,905
1,331 4.374 1.44 K.457 1.849 19,097 2.712 9.158 21143 12.764 2.545 21 115229 3.303 31 31.806
0L5V7 0.559 1,122 0,75H 4.788 1.184 13.5 1957 2.976 1.327 3.817 1.454 8644 1.67 21.518 2.87 4.477 1.581 5,469 1.89 9,07 2004 20.535 2.917 10.006 2.357 14.0% 2.741 12.122 3.482 33,864
2,946 18 18.696 3,283 28 235 3.974 39.844
43,548
2.481 11.534
MAE
3,281 21 21,864 3. 3.595 31 19.599 4,258
1.0 0.127 0.235 0.151 0.251 0.218 1.224 1.066 0.429 0.141 0.267 0.154 0.298 0223 0,339 0.959 0,513 0,215 0.386 0.257 0.393 0.304 0.111 0,893 0.516 2566 1.240 2.633 1.255 2.544 1.302
1.5
MSE MAE MSE
0.0 1,000 0,000 0,0000 0,000 0,000 1,000 0. 0,001 0,000) 0.000 0.000 0,000 0,000 0,000 0,0000 0,000 0,000 0,000 0,005 0.000 0,004 0,0000 0,004 0,000 O. 0.005 0,000 0,000 0,0010 0,000 0,000
7.000
9.856
Simplex
2.096
93369 2.429 9,461 2.400
2.0 0.040 0.159 (1941 16161 0.042 0.I61 0.048 0.171 0.04 1.167 0.046 0,166 0.049 0.171 0,089 O. 08186 0,042 O. 0.164 0.044 0.167 0.046 0,170 0,051 O. 1925 1.579 3,925
6132 1.974 6,133
6.47 2,024 6634
MSE MAE
2.0 1.427 0.890 2.549 1,09 6421 1.505 IS 15.1 2.174 7,089 2.092 8,339 2.228 13.968 2. 6998 26.111 3.592 8,456 2.272 9.608 2.374 14 250 2,738 25.467 3.486 14.838
3.807 17.938
MSE MAE
2.5 0.059 0.192 1.0611 (1, 195 0,661 1.195 0.069 0.2016 0.064 O. 1969 0,066 0.198 0.069 0,211 0.081 0,216 0,058 0. 0.194 0.001 0.197 MO6S 0,200 0.069 (1.219
0.5 0.054 0.152 0.064 0. 164 0.101 1,202 0,783 0.332 19 0. 0.252 0.114 0.277 0.144 0.271 0.717 0,402 0.161 0,295 O. 0. 0,303 O. 0.198 0,314 0.787 O. 0,400
0.264 08788 0.271 0.391 0.349 0.430 1.312 0.635 0.346 0.457 0.305 0.463 0.424 0.490 L,02 0.611
0.521 1.510 0.732 0.456 0.527 0.469 0.531 0,545 0.561 1.131 0.691
1,368 0.791
48
2.5 1.913 1,024 3.212 1,228 7,555 L,53 17. 7.266 2.338 9,441 2,425 10.667 2.571 17.055 3,066 30.380 383 3.915 11.328 2.699 12.511 2.748 17.449 3.104 28.804
MSE MAE MSE MAE
0.768 0.428 4.492 0.461 0,507 0,590 0,560 1.599 0.801 0.641 0.619 0.648 0,620 0,743 0.651
15
iTursformer
MSE MAE
L,ST 19.774 2.767 3,409
MSE MAE MSE MAE MSE MAE
48
1.5
0.5 0.255 0.364 0.797 11556 3642 LIMN 12,622 1823 2,108 1,069 2.797 1.19 7.024
0.630 0.357 0.461 0,380 0.472 0.494
Koopa
6 MSE MAE MSE MAE MSE MAE MSE MAD MSE MAE
0.540
48
1.5 0.222 0.323 0,360 0..42 0,572 0.384 1.520
2.0 0.303 0.385 0,342 0.402 0.478 0.446 1,540
1993
561
15
DeepEDD
0.504 0,736
2.5 4.455 0.481 0,519
Models
H
22

---

LETS Forecast: Learning Embedology for Time Series Forecasting
Table 12. Full ablation table
Danset
Linear
MLP
MLP+EDM
Full Model
Metre
MSE
MAE
MSE
MAE
MSE
MAB
MSE
MAE
48
0.1980
0,2704=0.om 0004
0.1880 200000
0.2673
0..091
0.2463:0.0002
0.160T=0.0011
0.2470=0.000
BCT.
0.15344 0002
0,24633 0003
0.1511 =0 0029
0.2478:0.0024
0.1380
0.2322=00000
0.1377 20.77200000
0.2317=00 0006
144
0.1526+a.0000
0,2461=0 0002
0,1531
0.2191=0.0019
0.14666a0.000
0.2393=0.000
0,1459
0.2392:0.00
192
0.1545510.000
0.248620
0.1533
0.2483
0.152820.00m
0.2458.0.000
0,1507 20.0000
0,2438
48
0.3412a0.0013
0.3630
0.3369
0.3654
0,3280
0.3599 0000
0.323620.001
0,3566
E7T12 BThh
96
0.3750
0.3866=0 0012
0.3712
0.3861
0.3687 2:
0.3873=00000
0,3622
0,3831 0004
144
0.3953 0009
0.4005 0008
0.3961
20.0011
0.4031 2000001
0.3981
2:0.004
0.4022-0:0019
0.3885
0.3977 0006
192
0. 4.096
0.4125 0013
s0.000
0..113+0.0015
a..180
0..261
0.404440.0074
0. 0.4213==0.0070
48
0.2258 0m03
0:2913:20 0002
0.2237
20.0009
0.2907 20.0003
1.22655a..002
1,2886=000010
0.2256.0.00m 0021
0.2875 0011
96
0.2838
0.3331 ::0 0002
0..00220.0002
20.0002
0.3310
20.0001
0,2935:00.000
0.3367=0.0025
0,288220.0015
0.3329
144
0.3202
0,3580 000a
00.061=0.0c2n
0.3546:0.0011
0.3317 #0.0022
0.3676=0.0018
1.3195=0.0000
0.3601 0006
0.3505
0.3787 0008
0..034
0.3745
0.355
0.3803=0.0088
0.3483=0.0090
0.3737
48
0.3045
0.3346 0008
0.3045=0.0004
0.33446:0.003
0.2766 20.0000
0.3178:000000
0.276820.0077
0.3184:0.0077
96
0.30-13+..001
0,3377= 0011
0.3035
0.3367
292 2921..0.0017
0.3305=0.0010
0,2885
0.3279
144
0.3218:40.002
0.3492 0001
0.3129
0.3470=0.0000
0.3106
0.3163=0.0010
0.3070
0.3437
192
0. 0.33644. 0007
0.3581 :0 0008
0.3282 0004
0.3583
0.3279 0227999.0077 0012
0.3566 0004
0.32 12 3.0.0000 00.00
0.3527 0011
48
0.14499 000e
0,23344=0
0000
0,1437 200000 0003
0. 00.335
0.1338 1006
0.2221=0.0m 0000
0.1330 0006
0.2210 D00G
B11m2
96
0.1718:+0.0mp 0003
0,2510=0.om 0001
00.707=0.0002
0. 0.2510=0.000
0.1710=0.0022
0. 0,2502=0.0022
0.1686
0.2478 0006
144
20.0004
0:2708 0001
0.0093.0.000m
M,26955.00000
0,2011 2000002
0.271 10271120000 0000077
0.202020.00m2
0.2714
192
0.2201+0.0002
0,2855 0002
0..092
0.2852.00.000
00.000
0.2908=0.0029
0.223420.0027
0.288G:000008
48
0.6978 0000
0.4131
0004
1.562120.000
0.3636 0.0000
0.4374:0.003
0,2863=0.001
0,4501 0008
0.2859 0011
Thilfc
0.4508
0.2985 0013
0.4241 0000
1.29744100002
0.3757
10.0015
0.2608_00001
0.382820.00m
0.2596
144
0..300
0.2912=0.or 0016
0.4112s0.00la
0.2871#0.0012
0.3781
60.0010
0.2615=000016
0.3504=6.0011
0.2582
192
0.421940.0m4
0,2876 0012
0.4111
00.00300.0000
0.3809
0,26441.
0.3870=0.001
0,2616
48
0.0430t00.00
0. 0.1428:: 0008
0.04223go.om 20.000
0.1409
0.0427
0.1422
0.0422 7.0422:0.000s
0.141120.0008
Exchinge
96
0,0841
0.2018 :0::8:0000 0006
0.0853=0.0019
0. 0.2029
0.0853#0.0014
0.2037=0.0010
0.0870=0.002
0,20411:0.003
144
0.1301 1:000000
0.2526 0002
0.1314220000
0.2550
0.0001
O. 0.1370 0092
0.2603=000000
0.131 920 0053
0.25 0.25446200009
192
0.1818:a0.001
0.3018 0017
0.18:4920.00f
0.3061
0..838
0.3053=0.00g 00gg3
0.1751 0000024
0.2989 0004
24
2.2286=a0.0111
0,9375 0040
2:2111=0.0017
18901=0.0006
1
0.8 193=0:0150
1.7489 1907
0.7896 0362
36
2.0815..0.00
0.9264 0003
1.935020..00
0.8443:20.010
1.5775
0.760620.00mt
1.640520.0700
0.769020.0157
1.8621 0090
0.9020 0011
1.6525 0007
0.8133
1.6184g0.074
0.7868
1.614920.0072
0.7880 0138
1.9258
0.9425 0589
2.1317=0.20 2682
0.9979a0.100
1.6955
0.8241 20.0200
1.7072=0.002
0.8274
0.066520.00122
0.1919::.om 0009
0.187022.000: 0.161.800.002
0.1396 0.0000
0.1705_00.002
2.03002000010
0.1683::0
Weather
96
0. 0.19381.0000
0.2218:10.o
0.1931
0.2196220.00c
0.1566
0.1917
0.1571
0001
0.1917
144
0. 0.2033
0.2341 0006
0.180320.0009
0.2214
0.1752
0.2105=0..oom
0.1739 14739900.00s
0.2093
192
0.21 22185:40.000 0005
0,2518 0014
0.2182=0.com 0006
0. 0.2512=0.000
0.1924:..u013
0,228 0.2281=00001
0.191320.0007
0.2263 000000
#lmprvements
55
52
54
Baseline
#Degradations
17
20
18
23


Reviews:
[{"review_id": "P1VtVbcyE2", "reviewer": "Reviewer_Ywcy", "summary": "This paper introduces a new framework, DeepEDm, for time series forecasting, inspired by studies in Empirical Dynamic Modeling (EDM). The approach is fundamentally based on Taken's theorem, which states that a dynamic system can be reconstructed using a delay embedding of the series in phase space.\n\nIn practice, the method starts by constructing a time-delay representation of the series, from which an embedding is learned in a latent space. Forecasting is then performed using a regression kernel, which assigns weights to the closest representations from the training set in the latent space. Finally, a classic MLP decoder generates the forecast.\n\nExperiments on both synthetic and real-world datasets demonstrate the performance of the proposed model.", "questions": "", "limitations": "", "rating": 3, "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": "Thank you for your thoughtful and detailed review. We now address your suggestions. Our response will be integrated into the final version.\n\n### 1. Clarification on Generalization Statement\n\nThis was meant to contrast DeepEDM and EDM w. simplex: EDM trains a separate model for each time series, whereas DeepEDM learns a single shared model across all series (within same dataset). The statement is partially supported by our main results (Table 1), where models were evaluated on unseen time series. In this widely-used evaluation protocol, the training series and test ones may be extracted from the same long sequence. Due to this design, we realize that part of this statement (“generalization to new time series”) is ambiguous and will revise it in the final version. \n\nTo further clarify, we conducted an additional experiment on ETT, Exchange, and Weather datasets to evaluate DeepEDM's ability to generalize across “different” time series (within the same dataset). Specifically, for each dataset, we trained the model on a subset of sequences (seqs.) and tested it on a different subset.  \n- For the ETT datasets, which consist of 7 seqs., we trained on the seq. 0-2 (using only timesteps from the standard training split) and tested on seq. 4-6 (using timesteps from the standard test split). We use this 3:3 split as several baseline models are unable to handle different channel dimensions between training and testing.\n- Similarly, for the Exchange dataset (8 seqs.), we trained on the first 4 sequences and tested on the latter 4. \n- For the Weather dataset (21 seqs.), we trained on sequences 0-9 and tested on sequences 10-19. \n\nWe compared DeepEDM against strong baselines including Koopa, iTransformer, and PatchTST. The prediction horizon (H) varied from [48, 96, 144, 192] and lookback window was set to $2*H$. The summarized results are presented below along with the link to full results. Notably, DeepEDM ranks 1st in MAE and MSE winning in 39 out of 48 cases. The results thus provide more direct support to the statement of “a single model that generalizes to new time series.\"\n\n| Dataset   | DeepEDM (MSE/MAE)     | Koopa (MSE/MAE)   | PatchTST (MSE/MAE)    | iTransformer (MSE/MAE)   |\n|:----------|:----------------------|:------------------|:----------------------|:-------------------------|\n| ETTh1     | (**0.230**/**0.310**) | (0.248/0.330)     | (0.253/0.330)         | (0.259/0.330)            |\n| ETTh2     | (**0.151**/**0.230**) | (0.170/0.260)     | (0.183/0.270)         | (0.170/0.250)            |\n| ETTm1     | (**0.214**/**0.280**) | (0.242/0.310)     | (0.216/0.290)         | (0.245/0.300)            |\n| ETTm2     | (**0.075**/**0.160**) | (0.086/0.180)     | (0.088/0.180)         | (0.096/0.190)            |\n| Exchange  | (0.106/**0.200**)     | (0.146/0.240)     | (**0.105**/0.210)     | (0.131/0.230)            |\n| Weather   | (**0.293**/**0.250**) | (0.325/0.280)     | (**0.293**/**0.250**) | (0.423/0.320)            |\n\nhttps://anonymous.4open.science/r/icml_rebuttal-B440/generalization_exp_results.md\n\n###   2. Lookback Window Reporting\n\nSince historical data is typically accessible, any reasonable lookback window can be assumed and the lookback length is often treated as a hyperparameter [Bergmeir, 2023]. Therefore, we tuned lookback length separately for each model and forecasting horizon, and only reported best results without a detailed breakdown. This protocol was also considered in prior works [TimeMixer, PatchTST, DLinear]. The effects of lookback length in forecasting performance was indeed discussed in Appendix A.5.2 (Figure 4). If needed, we can add a detailed breakdown to the Appendix. \n\n###  3. Ablation Study Reporting and Results of Linear Models\n\nOur ablation study is designed to concisely highlight the method design choices. These detailed experiments span 9 datasets, 4 lookback settings, and 4 configurations with multiple runs. To maintain clarity and conciseness, we presented aggregated results describing key trends. Per reviewer’s request, we have included the full table below, which we will add to the Appendix.\n\nhttps://anonymous.4open.science/r/icml_rebuttal-B440/ablation_full_results.md\n\n**Linear models**: \nAs prior work  [DLinear, RLinear] has noted, linear models are effective for forecasting, particularly for simpler datasets. However, they struggle with more complex datasets. As shown in the above table, DeepEDM significantly outperforms the linear model on benchmarks like ILI, Weather, ECL, and Traffic by a large margin, e.g., on ECL the relative improvement in MSE from the linear model to our method is ~10%.\n\n###  3. Transfer learning / domain adaptation\n\nThis is an exciting future direction! Our work focuses on single-domain forecasting but shows transfer within-domain (see Q1). Cross-domain transfer learning requires further study and we leave it as future work.", "claims_and_evidence": "The main claim is that the proposed method achieves strong performance compared to the state of the art and is theoretically grounded. The experimental campaign indeed demonstrates the effectiveness of the proposed approach.\nA secondary claim is mentioned in the introduction: \"it learns a single parametric model that generalizes to new time series.\" However, I am not entirely sure what the authors mean by this statement. My understanding might be incorrect, but I expected to see experiments on adaptation or transfer to previously unseen time series in the datasets. However, the paper does not include any experiments addressing this issue.", "methods_and_evaluation_criteria": "The selected benchmarks are standard in the field, although they are known for their simplicity and stability. The two evaluation metrics used—MAE and MSE—are widely used in forecasting. The chosen baselines are also comprehensive, ensuring broad coverage of existing architectures", "theoretical_claims": "The formalism is derived from Empirical Dynamic Modeling (EDM). It is clearly presented and easy to understand, with a summary of the mathematical tools provided in the appendix. The paper does not include any additional theoretical elements that require validation.", "experimental_designs_or_analyses": "The paper follows recent publications in terms of benchmark selection and evaluation protocols. However, the benchmarks commonly used in the field are known to be simple and highly stable.\n\nThe authors chose to present results in the main body of the paper based on a very long lookback window (twice the forecasting horizon), while results for other lookback windows are only provided in the appendix. However, these results are not detailed per lookback window; instead, only the best result is displayed (Table 3). I believe these results should be presented in greater detail, with a breakdown by lookback window size.\n\nAdditionally, Table 4 (ablation study) reports averaged results, which somewhat obscures key insights. Specifically, it shows that a simple linear model achieves results very close to both the baselines and the proposed model, on average, across all forecasting horizons. To better understand the challenge posed by the task and the added value of more complex methods over a simple linear model, these results should be detailed more thoroughly.\n\nIt is unfortunate that there are no experiments on transfer learning or domain adaptation. The proposed method seems well-suited for such tasks, given the learned latent space, and this could have been a key strength of the approach.", "relation_to_broader_scientific_literature": "The paper is in line with recent works in time series forecasting, but it shares the same limitations, namely the use of simple benchmarks that may no longer be suitable.\nThe paper also draws a connection between the framework used and self-attention, which is indeed very similar once the latent projection method is defined.", "essential_references_not_discussed": "NA", "other_strengths_and_weaknesses": "NA", "other_comments_or_suggestions": "As mentioned above, it would be beneficial to provide detailed experiments for the different lookback windows and the ablation study. Additionally, it would be valuable to include experiments on more competitive tasks.", "questions_for_authors": "NA", "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": "", "extra_scores": {}}, {"review_id": "sKxCWBJCxT", "reviewer": "Reviewer_TNFN", "summary": "The paper resorts to first principles, to examine the usefulness of using embedology (as in Takens' embedding theorem) in conjunction with neural networks. This is important and has been missing in the literature.", "questions": "", "limitations": "", "rating": 2, "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": "Thank you for your detailed feedback and recognition of our theoretical contributions. We appreciate your acknowledgment that embedology, despite its foundational role, has been largely overlooked and that our work helps bridge this gap by integrating Takens' embedding theorem with neural networks. Your recognition of our approach as a step toward extending EDM and providing theoretical insights into Transformer-based forecasting is also greatly valued. Below, we address your suggestions and will integrate these responses into the final version.\n\n###  1. Clarification on Figure 2B\n\nThe axis in Figure 2B is presented in absolute terms, which we will clarify in the final version. We chose to not normalize the data (e.g. mean subtraction, standardization) to preserve the raw performance of the model. While this increases error ranges—especially in chaotic Lorenz settings where noise sensitivity is high—it provides a more transparent evaluation. Detailed results are in Appendix A.7.\n\n###  2. Consistency of Empirical Analysis\n\nOur empirical analysis consists of experiments and results using (a) synthetic time series generated from known, challenging dynamic systems (Lorenz and Rossler); and (2) real-world time series from multiple application domains. For synthetic data, we demonstrated that DeepEDM is more robust to noise when compared to the vanilla EDM, and is more accurate when compared to other representative learning-based methods (Koopa and iTransformer). For real-world data, we showed that DeepEDM consistently outperforms prior methods across different settings (e.g., fixed vs. varying lookback length). This consistency was evidenced by the overall ranking of our method (e.g., in Table 1, DeepEDM ranks the top 1 for 36 out of 72 metrics while the next best model achieves 11) as well as the results of multiple runs (Table 5 in the Appendix). Additionally, we presented extensive ablation studies of our design and lookback length. We believe our current empirical analysis is sufficient to support the main claim of the paper. Yet we welcome specific suggestions to improve our empirical analysis.\n\n### 3. Sensitivity to lookback length\n\nTheoretically, Takens’ theorem puts constraints over the number of time-delayed steps (>2d for d-dimensional state space), yet does not speak to the number of samples needed (i.e., lookback length). Empirically,  we observe that the model exhibits minimal sensitivity to the lookback length beyond a certain threshold. Once the model has sufficient historical data, additional time steps—particularly those with similar dynamics—contribute little to improving forecast performance. On the other hand, too short a lookback window can degrade performance, as there may be insufficient information to generate an accurate forecast. This observation is shown in our paper: the existence of an optimal lookback length, or \"sweet spot,\" as illustrated in Figure 4.\n\n###  4. Moving Figure 4 to the main paper\n\nGiven the additional one-page allowance in the final version, we plan to incorporate this analysis into the main paper.\n\n###  5. Benefit of DeepEDM for Some Kinds of Time Series than for Others\n\nGreat question! We will include a discussion in the main paper. Our main findings from the empirical analysis are twofold. First, there is a sweet spot in the lookback length, which is dataset-dependent (discussed in our response #3). Second, DeepEDM is best suited to time series with a low to moderate number of input variates, as seen in datasets like the ETT set, Weather, ILI, and our simulated data. For datasets such as Traffic and ECL, which contain a larger number of variates, we observe a slight decline in performance. One possible reason is that DeepEDM considers a channel-wise prediction using a shared model across all channels, and thus assumes independence between input variates. With a growing number of variates, there is likely an increasing strength of correlation among these variates, which our DeepEDM can not effectively model. We plan to further explore this correlation among input variates in future work. \n\n###  6. Performance Under Noise\n\nWe highlight that noisy conditions present a significant challenge for forecasting models, leading to performance degradation in strong baselines such as iTransformer, Koopa, and the vanilla Simplex. However, DeepEDM demonstrates greater robustness, exhibiting less performance degradation compared to these baselines. As discussed in Sec 5.1 and further shown in Table 6, DeepEDM achieves the best performance among strong baselines in the presence of noise. These results support our claim on noise robustness, as also noted by reviewer XFih that “[DeepEDM] remains robust even in the presence of noisy observations”.", "claims_and_evidence": "DeepEDM claims three key advantages. \n(1) By learning a latent space of the time-delayed embeddings, it mitigates\nthe sensitivity of EDM to input noise; \n(2) Unlike EDM,which requires a separate model for each time series, it\nlearns a single parametric model that generalizes to new\ntime series; and\n (3) It offers flexibility and scalability, providing theoretic insights for Transformer-based\ntime series forecasting models", "methods_and_evaluation_criteria": "The integration of dynamical systems theory with time series forecasting is important and relatively under-explored.", "theoretical_claims": "The paper employs an efficient attention-based kernel regression, as a general framework to consider Transformer-based time series models. It maps time-delay embeddings, derived from Takens’ theorem, into a learnable, higher-dimensional latent space that is potentially noise-free, facilitating a more precise reconstruction of the system’s underlying dynamics. DeepEDM can be viewed as employing a modified cross-attention framework. Similar to cross attention, the focal vectors act as queries (Q), the historical data points (Y) serve as keys (K), while the future states (y + ∆t) act as the values (V). Further, the relationship between queries and keys is quantified using the dot product as the similarity metric modulated by θ. However, DeepEDM diverges from traditional attention mechanisms by enforcing a structural differentiation where keys and values are explicitly distinct.\n\nI am famialiar with the Takens type of embedding and in my opinion the approach in this paper is correct.", "experimental_designs_or_analyses": "The approach is evaluated on both synthetic and real world time series. The synthetic series involve the Lorenz and Rossler chaotic signals (which exhibit clear attractors).", "relation_to_broader_scientific_literature": "The paper extends and combines two classical approaches, within the framework of transformers:\n-  Empirical Dynamical Modeling (EDM) (Sugihara & May, 1990), \n-  Takens’ embedding theorem (Takens, 1981; Sauer et al., 1991),", "essential_references_not_discussed": "N/A", "other_strengths_and_weaknesses": "Strentghs: \n- the paper introduces a theoretical framework, but the exepriments do not fully justify the approach\n- The empirical results on real world data are good but not consistently outperforming SOTA. \n- the proposed method is not very sensitive to the lookback length (time segment size), which is unexpected\n\n\nWeaknesses: \n- are the curves in Fig 2plotted in the absolute terms or in a dB scale?\n- There is little consistency in the empirical analysis. \n- Is the proposed method better suited for some kinds of time series than for other kinds?\n- the simulations vs lookback length in Fig 4 in the supplement are important and better suited to be part of the main paper. \n- since the paper deals with multivatiate time series, I would have expected deep analysis of the performance wrt the number of variates in a multivariate time series\n- the performance under noise is not convincing", "other_comments_or_suggestions": "None", "questions_for_authors": "See Weaknesses", "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": "", "extra_scores": {"ethical_review_concerns": "None"}}, {"review_id": "hSa68ldBe2", "reviewer": "Reviewer_qg1c", "summary": "This paper introduces DeepEDM, a framework that integrates nonlinear dynamical systems modeling with deep neural networks for time series forecasting. Built on empirical dynamic modeling (EDM) and Takens' theorem, DeepEDM employs time-delayed embeddings in a latent space and uses kernel regression to approximate underlying dynamics. This approach aims to enhance forecasting accuracy by explicitly modeling complex time series dynamics.", "questions": "", "limitations": "", "rating": 3, "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": "Thank you for your valuable feedback and recognition of our paper’s clarity, structure, and experimental design. We appreciate the acknowledgment of our contribution to nonlinear dynamical systems through EDM-integrated deep learning. Below, we address your suggestions and will incorporate these responses into the final version.\n\n### 1. Noise Robustness of DeepEDM\n\nWe hypothesize that DeepEDM’s noise robustness stems from its learned kernel, which enables adaptive distance assessments for nearest-neighbor retrieval, rather than using the fixed Euclidean distance in traditional time-delay embeddings. Fixed metrics are more sensitive to perturbations, while the learned kernel adapts to the data, thus improving neighbor identification under noise.\n\nTo validate this, we conducted additional experiments (please refer to our response #1 to Reviewer XFih for more details) on nearest-neighbor retrieval. In short, we compare (i) the time-delay embedded space using Euclidean distance, and (ii) DeepEDM's latent space using the learned kernel for retrieving true neighbors in the state space using synthetic data with known dynamics. Our results show that the learned kernel in DeepEDM achieves higher recall under noise, confirming that the learned kernel indeed enhances robustness against noise. This robustness is further supported by the superior forecasting performance of DeepEDM compared to EDM w. Simplex, under noisy conditions (see Table 6).\n\n###  2. Sensitivity to Time Delay and Embedding Dimension\n\nWe conducted additional experiments to study the effects of time delay and embedding dimensions. Specifically, DeepEDM has two parameters that control time delay and embedding dimensions. First, $m$ defines the number of time-delayed steps, and $\\tau$ controls the interval between these steps. For example, $m=3$ and $\\tau=1$ results in a 3-dimensional embedding $[y_t, y_{t-1}, y_{t-2}]$, while $m=3$ and $\\tau=2$ would yield an embedding $[y_t, y_{t-2}, y_{t-4}]$. In our work, $m$ is determined empirically for each dataset, while $\\tau$ is set to 1. Following your suggestion, we conduct ablation experiments to analyze the impact of these hyperparameters. For the experiments on $m$, we fix $\\tau = 1$ and vary $m$ over $[3, 5, 7, 11, 15]$. Conversely, for the experiments on $\\tau$, we set $m = 5$ and explore $\\tau$ in $[1, 2, 3]$. The results are linked below.\n\n**Effects of time-delayed steps $m$ (i.e., embedding dimension)**: Our findings indicate that the impact of $m$ is dataset-dependent. In some cases, performance remains relatively stable across different values of $m$, suggesting that the intrinsic state-space dimensionality might either be very low or very high. If the state-space dimension is small, larger values of $m$ will capture the underlying dynamics and thus perform similarly well. On the other hand, if the state-space dimension is large, different small values of $m$ will not be able to model the dynamics, yielding results at a similar performance level. This behavior aligns with the intuition that effective embedding reconstruction depends on the underlying system complexity.  \n\nhttps://anonymous.4open.science/r/icml_rebuttal-B440/m_delay_results.md\n\n**Effects of delay interval $\\tau$:** Our results show that in most cases, $\\tau = 1$ yields the best performance, suggesting that a small stride is sufficient. This is also the case considered in Takens’ theorem. However, it remains possible that an optimal balance between $\\tau$ and $m$ could further improve results, which we leave for future investigation.  \n\nhttps://anonymous.4open.science/r/icml_rebuttal-B440/tau_results.md\n\n###  3. Performance degradation on High-Dimensional Datasets \n\nPlease refer to our response to Q5 from Reviewer TNFN.\n\n###  4. Loss Function Impact\n\nOur ablation study (Table 4) shows that our DeepEDM significantly enhances forecasting performance (MLP vs. MLP+EDM, without any loss function optimization) and the loss optimization further reduces the errors (MLP+EDM w/o TDT loss vs. Full Model w. TDT loss). To further delineate the contributions of DeepEDM and loss optimization, we conducted an additional experiment. Specifically, we compare the DeepEDM trained with standard MSE loss against DeepEDM trained with the optimization objective incorporated in our paper. The results of this experiment are available at the table linked below. Notably, the differences remain marginal—while in some cases, the MSE loss further improves DeepEDM’s performance, in others, it results in slight declines. \n\nhttps://anonymous.4open.science/r/icml_rebuttal-B440/loss_exp_results.md", "claims_and_evidence": "Overall, the claims made in the paper are clear.", "methods_and_evaluation_criteria": "Yes, the proposed methods and evaluation criteria are appropriate for the problem of time series forecasting.", "theoretical_claims": "The paper presents theoretical claims related to nonlinear dynamical systems and their integration with deep learning. While I have reviewed the theoretical justifications, I am not fully confident in the correctness of all derivations.", "experimental_designs_or_analyses": "I have reviewed the experimental design and analyses, and they appear to be sound.", "relation_to_broader_scientific_literature": "The paper contributes to the broader scientific literature by integrating nonlinear dynamical systems modeling with deep learning for time series forecasting.", "essential_references_not_discussed": "The paper appropriately discusses related work, and I did not identify any missing essential references.", "other_strengths_and_weaknesses": "Strengths  \n1. The paper is well-organized, making it easy to follow the motivation, methodology, and results.  \n2. The integration of empirical dynamic modeling (EDM) with deep learning for time series forecasting is novel and addresses the challenge of capturing nonlinear dynamics.  \n3. The proposed approach is supported by extensive experiments and analyses, demonstrating its effectiveness across multiple datasets.  \n\nWeaknesses  \n1. The paper claims that DeepEDM mitigates EDM's sensitivity to noise and is potentially noise-free. However, the underlying mechanism for this claim is not explicitly clear. Further clarification on how DeepEDM addresses this limitation would strengthen the argument.  \n2. The proposed method relies on the time delay hyperparameter, which is crucial in EDM-based approaches. How is the time delay selected in practice? Additional ablation studies on the effect of time delay and embedding dimension would be beneficial.  \n3. The method underperforms on certain datasets, particularly those with a high number of variables (e.g., Electricity, Traffic). Providing an explanation for this performance drop would improve transparency. Additionally, results using a more standard MSE-based objective would help verify that the reported performance gains are due to model improvements rather than loss function optimization.", "other_comments_or_suggestions": "N/A", "questions_for_authors": "Please refer to the Weaknesses.", "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": "", "extra_scores": {}}, {"review_id": "NuJ4CbcRYc", "reviewer": "Reviewer_XFih", "summary": "In this paper, the authors introduce DeepEDM, a framework that extends Empirical Dynamical Modeling (EDM) to learn complex latent non-linear dynamics from observed time series for improved forecasting. Building upon Takens’ theorem, DeepEDM first constructs time-delayed embeddings of the input data and then projects these embeddings into a high-dimensional latent space. Next, it employs kernel regression on this latent representation, and uses a multilayer perceptron (MLP) to transform the resulting predictions back to the original time-series space.\n\nAccording to the authors, DeepEDM offers three key advantages over existing methods:\n1. It remains robust even in the presence of noisy observations.\n2. It trains a unified parametric model that readily adapts to new time-series data.\n3. It combines the rigor of dynamical systems with the flexibility and scalability through its deep learning–based architecture.\n\nIn addition to describing fundamental concepts in empirical dynamical modeling and reviewing relevant literature on deep learning for time-series forecasting, the paper also draws connections between DeepEDM and the Attention mechanism. Extensive experiments on both simulated and real-world multivariate forecasting benchmarks show that DeepEDM performs competitively with state-of-the-art approaches, highlighting its potential for robust and adaptable time-series forecasting.", "questions": "", "limitations": "", "rating": 3, "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": "Thanks for your detailed feedback. We’re pleased that you recognize the novelty of combining Empirical Dynamical Modeling with deep learning and the strong performance of DeepEDM, especially in noisy settings. We also appreciate your acknowledgment of our connection to the Attention mechanism and the flexibility of our approach. Below, we address your questions.\n\n### 1. Validation of Latent Space Representation\n\nGreat question! The latent space in DeepEDM functions as a learned kernel. We hypothesize that with noise, the learned kernel retrieves nearest neighbors more faithfully in the underlying state space compared to the time-delay embedding, which, as dictated by Takens' theorem, relies on nearly noise-free data for accurate reconstruction. This is indirectly supported by our results in Table 6, where DeepEDM outperforms EDM in time series forecasting.\n\nTo provide a more direct validation, we design and conduct a new experiment using simulated time series generated from the Lorenz system, where true neighbors in the state space are known. We evaluate the ability to retrieve these neighbors using different distance metrics. Specifically, for a given state at time $t$, we first determine $k$ true neighbors in the original state space. For the same data point at time $t$, we then retrieve the top $m$ nearest neighbors in (i) the time-delay embedded space using Euclidean distance, and (ii) DeepEDM's latent space using the learned kernel. For each query point, we evaluate the recall of the retrieved neighbors, quantifying the fraction of true neighbors successfully retrieved. For a comprehensive evaluation across different settings, we conduct this experiment with varying values of $m \\in \\{7, 14, 28\\}$, fixing $k = 7$ (the minimum number of neighbors necessary for recovering the dynamics). We also consider two noise conditions: no noise ($\\sigma = 0$) and w. noise ($\\sigma = 2.5$). A high recall score indicates a distance metric can successfully identify true neighbors in the state space necessary to reconstruct the dynamics. The results of this experiment can be found here:\n\nhttps://anonymous.4open.science/r/icml_rebuttal-B440/nearest_neighbor_retrieval_exp_results.md\n\nAs expected, under noise-free conditions, both methods perform similarly. However, when noise is added, the vanilla time-delayed embeddings exhibit a sharp decline in recall. In contrast, our learned kernel in DeepEDM degrades more gracefully, maintaining more accurate retrieval of neighbors. This suggests that the state space reconstructed using the learned kernel more accurately reflects the underlying state space, as it preserves the local neighborhood structure.\n\n### 2. Generalization as a Unified Parametric Model\n\nPlease refer to our response #1 to reviewer Ywcy.\n\n### 3. Selection of baselines\n\nWe respectfully disagree with the comment, 'baselines (for synthetic dataset experiments) are predominantly EDM-based'. The following baselines were considered (for this experiment) in the paper:\n\n- **Simplex**: A classical EDM approach, as the starting point of DeepEDM.\n- **Koopa**: A recent deep model based on Koopman operator, as a representative method that combines deep learning and dynamical system modeling (Koopman).\n- **iTransformer**: A recent deep learning-based model, as a representative method that is pure learning based using a Transformer.\n\nThis selection ensures a well-rounded comparison against strong and methodologically diverse baselines. We note that the full set of results using these baselines is reported in Table 6 of the Appendix. \n\n### 4. Missing References on Chaotic Time Series Forecasting\n\nYour suggestion to incorporate prior work on chaotic time series forecasting is well taken. While our focus has been on EDM and deep learning-based forecasting, we recognize the relevance of research in chaotic time series forecasting. We drafted the following paragraph to add to our related work section. We welcome your further feedback.\n\n*A related research direction focuses on forecasting chaotic time series via state space reconstruction, mirroring the underlying principles of EDM. Pioneering work by Farmer & Sidorowich (Phys. Rev. Lett. 1987) introduced local approximation techniques within reconstructed state spaces using delay embeddings, facilitating short-term predictions. Subsequent studies explored the application of feedforward neural networks for learning direct mappings from reconstructed phase states to future states (Karunasinghe & Liong Journal of Hydrology 2006). Recurrent neural networks, particularly Echo State Networks (ESNs), have also shown promise, with adaptations like robust ESNs (Li, Hand & Wang. IEEE TNNLS 2012) addressing the inherent sensitivity of chaotic signals to noise and outliers. However, a significant gap remains: the development of a fully differentiable, end-to-end trainable neural network architecture that seamlessly integrates dynamical systems theory with deep learning methodologies.*", "claims_and_evidence": "Authors claim that DeepEDM:\n1. remains robust even in the presence of noisy observations.\n2. trains a unified parametric model that readily adapts to new time-series data.\n3. combines the rigor of dynamical systems with the flexibility and scalability through its deep learning–based architecture.\n\nOverall, the paper effectively supports its first claim—that DeepEDM remains robust in the presence of noisy observations, by demonstrating strong performance in both synthetic noise-injected scenarios and on real-world datasets against various state-of-the-art models. \n\nThe second claim, involving a “unified parametric model” that readily adapts to new time-series data, is the least substantiated: after introducing the idea, the paper does not provide further elaboration or experiments to confirm how DeepEDM generalizes to new time series. Consequently, while the framework appears promising, further clarity is needed on the role and validation of the latent dynamics, as well as concrete evidence of adaptation to new data.\n\nIt also partially supports its third claim of merging dynamical systems rigor with deep learning flexibility, as the authors present a novel architecture grounded in Takens’ theorem and empirical dynamical modeling. While the overall design is innovative, key details are missing about the latent-projection step. Specifically, it is unclear whether the “latent dynamics” it learns genuinely reflect the underlying system or directly improve the forecast. The paper does not delve into interpretability or show whether these latent dynamics match “true” hidden states in controlled scenarios (beyond broad performance metrics).", "methods_and_evaluation_criteria": "Yes, the paper’s methods and chosen benchmarks do make sense for time-series forecasting, as it uses standard datasets and builds upon widely studied dynamical systems for its simulations. However, the limited scope of comparisons in the simulated scenarios—focusing mostly on EDM-based approaches such as Simplex, Koopa, and iTransformers—raises some concerns. On the other hand, their comparisons on standard benchmark datasets are reasonably thorough, which partly compensates for the narrower set of baselines in the simulation experiments.", "theoretical_claims": "There are no proofs to verify beyond the statements of Takens’ theorem, which the authors cite rather than prove themselves.", "experimental_designs_or_analyses": "In the simulation study, authors inject Gaussian noise into well-known chaotic systems (e.g., Lorenz, Rössler) to test the model’s robustness; this is a reasonable to demonstrate DeepEDM's resilience to noise. However, as previously noted, it only compares DeepEDM with a limited set of methods (mostly EDM-based). This narrow comparison leaves open the question of how DeepEDM would fare against a broader set of state-of-the-art models in similarly controlled, noise-injected scenarios.", "relation_to_broader_scientific_literature": "The paper appears to be the first to systematically merge Empirical Dynamical Modeling with deep neural architectures for time-series forecasting. While prior work has explored EDM (e.g., Simplex, S-Map) and separate lines of research have focused on deep learning (e.g., Transformers, MLP-based approaches), no previous study has explicitly integrated these two strands under a single, end-to-end framework.", "essential_references_not_discussed": "The paper covers most of the relevant literature on empirical dynamical modeling (EDM) and deep time-series forecasting. However, despite DeepEDM’s direct connection to chaotic time series prediction, the authors do not provide a literature review of this field. Foundational work, such as Farmer & Sidorowich (1987), established key principles for short-term chaotic forecasting using time-delay embeddings, which are directly relevant to DeepEDM’s methodology. More recent studies have extended these ideas using neural networks, including:\n\n1. Karunasinghe, D. S., & Liong, S. Y. (2006). Chaotic time series prediction with a global model: Artificial neural network. Journal of Hydrology, 323(1-4), 92-105.\n2. Li, Decai, Min Han, and Jun Wang. \"Chaotic time series prediction based on a novel robust echo state network.\" IEEE Transactions on Neural Networks and Learning Systems 23.5 (2012): 787-799.\n\nIncluding a discussion of this literature would better position DeepEDM in the context of prior neural-based approaches to chaotic forecasting and clarify its novelty in comparison to existing methods.", "other_strengths_and_weaknesses": "The paper presents a novel combination of Empirical Dynamical Modeling (EDM) and deep learning, which is an original and promising direction for time-series forecasting. However, one key concern is that the model’s performance is primarily evaluated against standard forecasting benchmarks, which may not fully demonstrate its advantage in capturing underlying system dynamics. While these benchmarks are widely used in time-series forecasting, they are not necessarily designed to evaluate methods that explicitly model chaotic or nonlinear dynamical systems. A more extensive comparison against known chaotic time series datasets, beyond the current synthetic experiments in the simulation study, would help clarify whether DeepEDM provides meaningful improvements over deep learning-based methods specifically designed for chaotic time-series prediction.", "other_comments_or_suggestions": "I have no additional comments.", "questions_for_authors": "1. How do you validate that the latent space learned by DeepEDM accurately captures the underlying system dynamics?\nThe paper presents the latent projection as a core component, but it is unclear whether it is reconstructing meaningful system dynamics or simply acting as a high-dimensional feature extractor.\n\n2. Can you clarify what is meant by DeepEDM being a “unified parametric model” that generalizes to new time series?\nThe introduction suggests that DeepEDM can adapt to new time series data more effectively than EDM, but this claim is not well-supported in the rest of the paper.\n\n3. Why does the paper not reference prior work on chaotic time series forecasting, despite its direct relevance?", "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": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper introduces DeepEDM, a framework that integrates nonlinear dynamical systems modeling with deep neural networks for time series forecasting. The authors have addressed the concerns from the reviewers. Most of the reviewers support the paper. Please incorporate all the discussions into the final version of the paper.
Decision: Accept (poster)