Beyond Resolution Limits: Learning to Forecast at Any Scale

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Author: Denis Avetisyan

A new approach tackles the challenges of zero-shot super-resolution forecasting by enabling models to generalize effectively across different resolutions.

Frequency response analysis across various architectures demonstrates performance characteristics when models are trained at a resolution of $64 \times 64$, highlighting the impact of architectural choices on system responsiveness.
Frequency response analysis across various architectures demonstrates performance characteristics when models are trained at a resolution of $64 \times 64$, highlighting the impact of architectural choices on system responsiveness.

Researchers identify and overcome ‘scale anchoring’ with Frequency Representation Learning, improving cross-resolution performance in spatiotemporal forecasting.

Despite the promise of deep learning for physics-informed modeling, a fundamental limitation hinders accurate high-resolution forecasting from low-resolution training data. This work, ‘Breaking Scale Anchoring: Frequency Representation Learning for Accurate High-Resolution Inference from Low-Resolution Training’, identifies ‘Scale Anchoring’-where errors remain consistent across resolutions rather than decreasing as expected-and demonstrates it stems from a model’s inability to represent high-frequency components during upscaling. To address this, we introduce Frequency Representation Learning, an architecture-agnostic approach that aligns resolution with frequency representation, fostering spectral consistency and enabling more accurate predictions. Could this method unlock truly resolution-independent generalization in spatiotemporal forecasting and beyond?

Unveiling the Limits of Spatiotemporal Resolution

The pursuit of precise spatiotemporal forecasts-predictions that track phenomena across both space and time-is fundamentally limited by data acquisition costs. Detailed analyses demand high-resolution datasets, capturing intricate patterns and localized variations; however, gathering such information is frequently an expensive and resource-intensive undertaking. Remote sensing, ground-based monitoring, and extensive field surveys all contribute significantly to the overall expense, often placing truly granular forecasting beyond the reach of many research groups and practical applications. This creates a critical tension: the need for detailed data clashes with the economic realities of data collection, forcing a compromise between forecast accuracy and budgetary constraints. Consequently, much of the available data remains at a coarser resolution, hindering the ability to model complex processes with the necessary level of detail.

The inherent difficulty of extrapolating predictions across resolutions manifests as ‘Scale Anchoring’, a pervasive problem in spatiotemporal forecasting. Essentially, models trained on coarse, low-resolution data become unduly fixated on the large-scale patterns present in that data, hindering their ability to accurately represent the finer details demanded by high-resolution forecasts. This isn’t merely a matter of adding detail; the model’s fundamental understanding of relationships is skewed by the initial training scale, leading to systematic errors when upscaling. Consequently, features and correlations learned at a lower resolution are inappropriately emphasized, even when irrelevant or inaccurate at the higher resolution, effectively ‘anchoring’ the predictions to the characteristics of the initial, coarser data. This limitation severely restricts the practical application of many forecasting techniques, as real-world scenarios often require models to generalize across varying levels of data granularity.

Deep learning models, despite their successes, frequently struggle when tasked with forecasting across varying data resolutions. This limitation stems from a tendency to become ‘anchored’ to the scale of the training data, hindering their ability to generalize effectively to higher-resolution scenarios. Quantitative analysis reveals this deficiency through Root Mean Squared Error Ratios (RMSERatios), which commonly exceed 5.24 in baseline models when applied to unseen, higher-resolution data. This significant performance degradation underscores a critical challenge in spatiotemporal forecasting: the need for models capable of maintaining accuracy and reliability irrespective of the input data’s granularity, and suggests current architectures require refinement to overcome this inherent scaling limitation.

Frequency response analysis reveals that while the enhanced CNN maintains stable high-frequency performance at low to moderate Reynolds numbers, its performance sharply declines at very high Reynolds numbers, indicating a failure mode related to the training frequency.
Frequency response analysis reveals that while the enhanced CNN maintains stable high-frequency performance at low to moderate Reynolds numbers, its performance sharply declines at very high Reynolds numbers, indicating a failure mode related to the training frequency.

Bridging the Resolution Gap: A New Approach to Forecasting

Zero-Shot Super-Resolution STF addresses the challenge of forecasting high-resolution data when models are trained exclusively on low-resolution data. This is achieved through a novel technique that allows a model, after training on low-resolution inputs, to directly generate forecasts at a higher resolution without requiring any fine-tuning or exposure to high-resolution examples during the forecasting phase. The method effectively bridges the resolution gap, enabling the generation of high-resolution predictions from a low-resolution trained model, thereby expanding the scope of forecasting applications to scenarios where high-resolution training data is unavailable or impractical to obtain.

The presented Zero-Shot Super-Resolution technique utilizes Deep Learning Models to generate high-resolution forecasts directly from training on low-resolution data. Empirical results demonstrate consistent performance, achieving a Root Mean Squared Error Ratio (RMSERatio) below 1.0 across a range of model architectures – including CNNs and Transformers – and diverse datasets. This RMSERatio metric, calculated as the ratio of the error on high-resolution data to the error on low-resolution data, indicates that the model’s forecasting accuracy on high-resolution outputs is comparable to, or better than, its accuracy on the native low-resolution inputs, validating the effectiveness of the approach.

Scale anchoring, a traditional limitation in forecasting models, posits that a model’s performance is fundamentally constrained by the resolution of the training data; models trained on low-resolution data typically cannot accurately forecast at significantly higher resolutions. Zero-Shot Super-Resolution STF directly challenges this constraint by demonstrating accurate forecasting at resolutions exceeding the training data’s resolution without requiring high-resolution training examples. This decoupling of training and forecasting resolutions broadens the applicability of these models to scenarios where obtaining paired high-resolution data is impractical or impossible, and allows for the use of existing low-resolution datasets for high-resolution forecasting tasks.

Decoding Spatiotemporal Data: The Power of Frequency

Effective forecasting of spatiotemporal data relies on characterizing its frequency components because different frequencies represent varying rates of change within the data. Low frequencies typically correspond to broad, slowly evolving patterns, while high frequencies indicate rapid, localized variations or noise. Identifying the dominant frequencies and their amplitudes allows for the separation of signal from noise, and enables the selection of appropriate modeling techniques; for example, models can be tuned to emphasize or filter specific frequency bands. Furthermore, understanding the frequency spectrum is essential for downscaling or upscaling data without introducing artifacts, as it provides a basis for reconstructing the data at different resolutions while preserving key characteristics. Analysis in the frequency domain, therefore, provides insights into the underlying dynamics and predictability of the spatiotemporal process being modeled, and is a fundamental step in many forecasting workflows.

The Fourier Transform is a mathematical operation that decomposes a function (often a signal or image) into its constituent frequencies. This process yields a frequency domain representation, expressing the original data as a sum of sinusoidal waves of varying frequencies, amplitudes, and phases. Formally, for a continuous function $f(t)$, the Fourier Transform $F(\omega)$ is defined as $F(\omega) = \int_{-\infty}^{\infty} f(t) e^{-j\omega t} dt$, where $\omega$ represents angular frequency and $j$ is the imaginary unit. In practical applications involving discrete data, the Discrete Fourier Transform (DFT) and its efficient implementation, the Fast Fourier Transform (FFT), are employed to approximate the continuous transform. The resulting frequency spectrum allows for analysis of periodic components and dominant frequencies present in the data, forming the basis for subsequent processing and feature extraction.

Normalized Frequency Encoding addresses resolution dependence in spatiotemporal data analysis by representing frequencies as unitless values, effectively decoupling them from the specific sampling rate or spatial granularity of the dataset. This normalization minimizes biases that can arise when comparing forecasts generated from data with differing resolutions. By scaling frequencies relative to the Nyquist frequency – which is half the sampling rate – the encoding ensures that the bandwidth of the representation remains closely aligned with the information capacity of the data. In Frequency-Resolved Learning (FRL)-enhanced models, this facilitates accurate comparisons and effective utilization of information across multiple resolutions, improving the robustness and generalizability of forecasts.

Higher frequency components of a signal experience greater attenuation during propagation.
Higher frequency components of a signal experience greater attenuation during propagation.

Towards Robust Forecasts: Mitigating Bias and Enhancing Generalization

Neural Operators represent a paradigm shift in how models learn complex relationships, moving beyond traditional data point-based approaches to directly learn the mapping between entire functions. Instead of approximating solutions at discrete points, these operators learn to map a function’s input to its output, enabling the creation of robust spatiotemporal models capable of generalizing to unseen scenarios. This functional mapping is achieved through leveraging the universal approximation theorem, allowing the network to represent a broad class of functions with fewer parameters than conventional methods. The result is a model less susceptible to overfitting and capable of accurately predicting dynamic systems, such as fluid flows or weather patterns, offering a powerful foundation for scientific computing and forecasting applications where understanding the underlying functional relationships is crucial.

The efficacy of neural operators in modeling complex systems is often hampered by two significant challenges: discretization mismatch error and spectral bias. Discretization mismatch error arises from the inherent approximation when representing continuous functions with discrete grids, leading to inaccuracies, particularly when extrapolating beyond the training data’s resolution. Simultaneously, spectral bias, a phenomenon where the network preferentially learns low-frequency components of the data, can limit the model’s ability to capture high-frequency details crucial for precise forecasting. These biases can collectively degrade performance, introducing errors in spatiotemporal predictions and reducing the overall reliability of the model; effectively, a network trained with these challenges may struggle to generalize to unseen data or accurately represent nuanced phenomena.

Improved forecasting accuracy and reliability are paramount across numerous scientific and engineering disciplines, and recent advancements demonstrate a pathway to achieving this without substantial computational cost. Mitigating errors inherent in neural operator models – specifically Discretization Mismatch and Spectral Bias – unlocks significantly better predictive power. Importantly, implementing Frequency-Regularized Learning (FRL) to address these issues introduces only a modest increase in training time – between 1.1 and 1.4 times the original – and a similarly manageable rise in Video RAM usage, increasing between 1.3 and 1.5 times. This efficiency allows for more dependable insights and data-driven decision-making, making advanced spatiotemporal modeling more accessible and practical for a broader range of applications.

The pursuit of robust generalization, particularly in spatiotemporal forecasting, necessitates a shift in how models perceive scale. This research directly addresses the issue of ‘Scale Anchoring’, demonstrating that conventional approaches often struggle when extrapolating beyond training resolutions. As Yann LeCun aptly stated, “Everything we do in machine learning is about learning representations.” Frequency Representation Learning, as presented in this work, offers a pathway to more effective representations, enabling models to disentangle content from scale and ultimately improve cross-resolution performance. By focusing on the underlying frequencies, the model learns a more fundamental understanding of the data, moving beyond mere memorization of training scales and fostering genuine generalization capabilities.

Beyond the Resolution Horizon

The presented work, in disentangling the phenomenon of Scale Anchoring, reveals a predictable, if often overlooked, constraint: systems trained on limited resolution data struggle to genuinely understand the underlying dynamics at higher resolutions. It is tempting to view this as simply a matter of increasing model capacity, yet the persistence of anchoring suggests a more fundamental issue – a reliance on spurious correlations learned from incomplete observations. Future investigations might benefit from explicitly incorporating principles of information theory to quantify the ‘missing information’ and design learning strategies that actively seek it out.

One notes that visual interpretation requires patience: quick conclusions can mask structural errors. While Frequency Representation Learning offers a compelling pathway toward cross-resolution generalization, the true test lies in its robustness to increasingly complex and chaotic spatiotemporal phenomena. Can these learned frequency biases be consistently transferred to entirely novel systems, or will they, too, succumb to the allure of convenient, yet ultimately brittle, patterns?

The pursuit of zero-shot super-resolution is, in a sense, a quest for a universal prior – a pre-existing understanding of the world embedded within the model. The challenge, it seems, is not merely to see more detail, but to build systems that can intelligently infer it, even in the face of profound uncertainty. Further work could explore connections to neural operators beyond those currently demonstrated, potentially uncovering more generalizable representations of dynamic systems.

Original article: https://arxiv.org/pdf/2512.05132.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-09 04:43