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1. Ensemble bootstrap methodology for forecasting dynamic growth processes using differential equations: application to epidemic outbreaks

   https://doi.org/10.1186/s12874-021-01226-9  

   Chowell, Gerardo; Luo, Ruiyan (December 2021, BMC Medical Research Methodology)

   null (Ed.)

   Abstract Background Ensemble modeling aims to boost the forecasting performance by systematically integrating the predictive accuracy across individual models. Here we introduce a simple-yet-powerful ensemble methodology for forecasting the trajectory of dynamic growth processes that are defined by a system of non-linear differential equations with applications to infectious disease spread. Methods We propose and assess the performance of two ensemble modeling schemes with different parametric bootstrapping procedures for trajectory forecasting and uncertainty quantification. Specifically, we conduct sequential probabilistic forecasts to evaluate their forecasting performance using simple dynamical growth models with good track records including the Richards model, the generalized-logistic growth model, and the Gompertz model. We first test and verify the functionality of the method using simulated data from phenomenological models and a mechanistic transmission model. Next, the performance of the method is demonstrated using a diversity of epidemic datasets including scenario outbreak data of the Ebola Forecasting Challenge and real-world epidemic data outbreaks of including influenza, plague, Zika, and COVID-19. Results We found that the ensemble method that randomly selects a model from the set of individual models for each time point of the trajectory of the epidemic frequently outcompeted the individual models as well as an alternative ensemble method based on the weighted combination of the individual models and yields broader and more realistic uncertainty bounds for the trajectory envelope, achieving not only better coverage rate of the 95% prediction interval but also improved mean interval scores across a diversity of epidemic datasets. Conclusion Our new methodology for ensemble forecasting outcompete component models and an alternative ensemble model that differ in how the variance is evaluated for the generation of the prediction intervals of the forecasts. 

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2. Diffusion Probabilistic Modeling for Video Generation

   https://doi.org/10.3390/e25101469  

   Yang, Ruihan; Srivastava, Prakhar; Mandt, Stephan (October 2023, Entropy)

   Denoising diffusion probabilistic models are a promising new class of generative models that mark a milestone in high-quality image generation. This paper showcases their ability to sequentially generate video, surpassing prior methods in perceptual and probabilistic forecasting metrics. We propose an autoregressive, end-to-end optimized video diffusion model inspired by recent advances in neural video compression. The model successively generates future frames by correcting a deterministic next-frame prediction using a stochastic residual generated by an inverse diffusion process. We compare this approach against six baselines on four datasets involving natural and simulation-based videos. We find significant improvements in terms of perceptual quality and probabilistic frame forecasting ability for all datasets. 

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3. SDYN-GANs: Adversarial learning methods for multistep generative models for general order stochastic dynamics

   https://doi.org/10.1016/j.jcp.2024.113442  

   Stinis, P; Daskalakis, C; Atzberger, PJ (December 2024, Journal of Computational Physics)

   We introduce adversarial learning methods for data-driven generative modeling of dynamics of n-th-order stochastic systems. Our approach builds on Generative Adversarial Networks (GANs) with generative model classes based on stable m-step stochastic numerical integrators. From observations of trajectory samples, we introduce methods for learning long-time predictors and stable representations of the dynamics. Our approaches use discriminators based on Maximum Mean Discrepancy (MMD), training protocols using both conditional and marginal distributions, and methods for learning dynamic responses over different time-scales. We show how our approaches can be used for modeling physical systems to learn force-laws, damping coefficients, and noise-related parameters. Our adversarial learning approaches provide methods for obtaining stable generative models for dynamic tasks including long-time prediction and developing simulations for stochastic systems. 

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4. Learning Residual Distributions with Diffusion Models for Probabilistic Wind Power Forecasting

   https://doi.org/10.3390/en18164226  

   Chen, Fuhao; Gao, Linyue (August 2025, Energies)

   Accurate and uncertainty-aware wind power forecasting is essential for reliable and cost-effective power system operations. This paper presents a novel probabilistic forecasting framework based on diffusion probabilistic models. We adopted a two-stage modeling strategy—a deterministic predictor first generates baseline forecasts, and a conditional diffusion model then learns the distribution of residual errors. Such a two-stage decoupling strategy improves learning efficiency and sharpens uncertainty estimation. We employed the elucidated diffusion model (EDM) to enable flexible noise control and enhance calibration, stability, and expressiveness. For the generative backbone, we introduced a time-series-specific diffusion Transformer (TimeDiT) that incorporates modular conditioning to separately fuse numerical weather prediction (NWP) inputs, noise, and temporal features. The proposed method was evaluated using the public database from ten wind farms in the Global Energy Forecasting Competition 2014 (GEFCom2014). We further compared our approach with two popular baseline models, i.e., a distribution parameter regression model and a generative adversarial network (GAN)-based model. Results showed that our method consistently achieves superior performance in both deterministic metrics and probabilistic accuracy, offering better forecast calibration and sharper distributions. 

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5. Reconstructing Galaxy Cluster Mass Maps using Score-based Generative Modeling

   https://doi.org/10.33232/001c.142147  

   Hsu, Alan; Ho, Matthew; Lin, Joyce; Markey, Carleen; Ntampaka, Michelle; Trac, Hy; Póczos, Barnabás (July 2025, The Open Journal of Astrophysics)

   We present a novel approach to reconstruct gas and dark matter projected density maps of galaxy clusters using score-based generative modeling. Our diffusion model takes in mock SZ and X-ray images as conditional inputs, and generates realizations of corresponding gas and dark matter maps by sampling from a learned data posterior. We train and validate the performance of our model by using mock data from a hydrodynamical cosmological simulation. The model accurately reconstructs both the mean and spread of the radial density profiles in the spatial domain, indicating that the model is able to distinguish between clusters of different mass sizes. In the spectral domain, the model achieves close-to-unity values for the bias and cross-correlation coefficients, indicating that the model can accurately probe cluster structures on both large and small scales. Our experiments demonstrate the ability of score models to learn a strong, nonlinear, and unbiased mapping between input observables and fundamental density distributions of galaxy clusters. These diffusion models can be further fine-tuned and generalized to not only take in additional observables as inputs, but also real observations and predict unknown density distributions of galaxy clusters. 

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