A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems

arXiv:2603.25370 · stat.ML, cs.LG · Submitted 2026-03-26 · Read on arXiv

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Introduction to the show: ident: AI Radio. Generated commentary on the latest Artificial Intelligence papers.

Tom: Today's paper: "A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems".

Jane: The gist A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions,

Tom: First, who's behind it and why it matters.

Title and authors: Tom: So we're looking at this paper today, "A Distribution-to-Distribution Neural Probabilistic Forecasting Framework for Dynamical Systems." It sounds heavy, but basically, they’re trying to move beyond just predicting one single future state.

Jane: Right. So the core idea is treating the prediction itself as a distribution instead of just a point. That means uncertainty isn't something you add on at the end, it's built right into how the model evolves over time, which is pretty fundamental for systems that are changing.

Lu: The authors, Yang, Du, and Aslett—they are tackling this by making these predictive distributions first-class computational objects in their neural network setup. They’re not just using them as inputs or outputs; they're making them the actual data the system processes.

Meng: From a deployment standpoint, if you can operate directly on distributions, it suggests a cleaner way to handle things than having to run massive ensembles every time you want a forecast. It could simplify the pipeline significantly if that works out in practice.

Lalam: I'm seeing that this paper focuses specifically on marginal predictive distributions for each state dimension, instead of trying to model the whole joint distribution at once, which keeps the parameterization manageable for now.

Tom: Exactly, it’s about focusing on those individual pieces—like how fast one variable is changing—and then feeding those representations into a neural network that learns how they evolve together over several time steps.

The paper's summary: Jane: So what does this actually propose? Essentially, they introduce a distribution-to-distribution framework where the input and output of the forecasting model are both predictive distributions. That allows for a recursive propagation of uncertainty across successive forecast steps.

Lu: They define this by having an encoder that takes your input distributions and maps them into a finite-dimensional representation suitable for neural processing, then a neural forecasting module, and finally a decoder that turns those representations back into parameterised predictive distributions.

Tom: I like how they frame it modularly. You can swap out the core forecasting module—like using an LSTM or a Transformer—while keeping the distribution encoder and decoder structure intact, which is good for experimentation.

Meng: The summary mentions that this avoids the common issue where uncertainty is just an auxiliary quantity tacked onto a deterministic forecast. They are trying to make uncertainty intrinsic to the model's evolution itself, which I think is important for real-world reliability when things get noisy.

Lalam: They use kernel mean embedding, which they explain as mapping input distributions into a Reproducing Kernel Hilbert Space. This gives them a principled way to represent those inputs in a finite-dimensional space for the neural network to handle easily.

Jane: So it’s taking something continuous and probabilistic—like the forecast for tomorrow—and converting it into discrete vectors that the standard neural architecture can chew on, and then decoding it back into another probabilistic forecast.

The paper's improvements: Tom: One of the main improvements they highlight is shifting from trajectory-oriented methods to a direct approach where predictive distributions evolve dynamically through time. That’s what page three shows, pt plus k equals F(k) Theta (pt) <ref:2603.25370#pg1>.

Lu: That iterative structure is key because it lets the model learn a dynamic distributional evolution operator. It’s not just predicting the next step based on the current state; it's learning a rule for how distributions themselves transform over time.

Meng: What I find interesting, from an implementation standpoint, is that this allows for training driven by probabilistic performance metrics, specifically minimizing things like the empirical logarithmic score. They aren't just optimizing for a single next step; they are optimizing the whole sequence against how well those sequences match observations across many samples.

Jane: That leads to another big improvement: the framework is trained explicitly by these probabilistic performance metrics. They use that objective function to drive the learning process, which means the model learns to produce distributions that are actually skillful at predicting what we see in reality.

Lalam: The way they parameterize their output distributions using Gaussian mixture networks, specifically M=five components for each of the three state variables, allows them to capture non-Gaussian and multimodal predictive marginal distributions very flexibly.

Conclusion: Tom: So to wrap up, the main point of this paper is that by operating directly on predictive distributions instead of samples, you can create a unified end-to-end architecture for forecasting where uncertainty is an intrinsic part of the system's evolution.

Jane: That means we move away from treating uncertainty as a side effect and start learning an operator that produces skillful predictive marginal distributions consistent with the underlying dynamics. It’s about making the model's uncertainty a core feature, not just noise.

Lu: The iterative strategy of evolving those marginal distributions dynamically over time is what makes this structure robust for multi-step forecasting, as it avoids issues where you apply a model to a distribution that was never seen during training.

Meng: For us in production, the flexibility of using mixture density networks to output flexible distributions is attractive because it lets us represent complex predictive scenarios without needing impossibly large ensembles or complex sampling routines for every single forecast.

Lalam: I’m confident that this framework provides a principled way to map those marginal distributions into finite-dimensional neural representations using kernel mean embeddings, which keeps everything structurally compatible with the Gaussian mixture outputs they are aiming for.

Tom: It’s a solid piece of work because it connects representation learning techniques—like KME—with probabilistic modeling in a way that seems computationally tractable.

Jane: Yeah, it’s certainly something worth watching as researchers continue to push these types of models toward more complex, real-world dynamical systems. That's where the next step is headed.

Department of Mathematical Sciences, Durham University · School of Mathematics, East China University of Science and Technology · Data Science Institute, The London School of Economics and Political Science

stat.ML, cs.LG

Submitted: 2026-03-26

Updated: 2026-10-14

Comments: 11 pages,5 figures

License: http://creativecommons.org/licenses/by/4.0/

Importance score: 80/100

The gist: The gist A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions, enabling recursive propagation of uncertainty

Key concepts

Distribution-to-Distribution (D2D) Framework: This is a novel approach where the entire forecasting process—input encoding and output decoding—operates directly on probability distributions rather than single predicted values. It treats uncertainty as a primary object to be propagated through successive forecast steps, ensuring that the model maintains a coherent representation of prediction uncertainty throughout its operation.
Kernel Mean Embedding (KME): KME is a technique used to map complex input distributions into a finite-dimensional vector suitable for neural networks. It uses a positive definite kernel function to represent the distribution in a Reproducing Kernel Hilbert Space (RKHS). This embedding allows the model to process the distribution's shape efficiently while preserving crucial properties like linearity with respect to distribution mixing.
Mixture Density Networks (MDN): MDNs are used as the output mechanism of the framework. They parameterize predictive distributions by modeling them as a Gaussian mixture, meaning they can capture complex, non-Gaussian, and multimodal uncertainty in the forecast. This flexible output form is compatible with the encoding mechanism, ensuring that input and output distributions share a consistent functional structure.

Terminology

Uncertainty Quantification — 16× in this paper · explained in 9 episodes
Probabilistic Forecasting — 15× in this paper · explained in 3 episodes
Embeddings — 10× in this paper · explained in 2 episodes
Decoder — 9× in this paper · explained in 2 episodes
Benchmark — 6× in this paper · explained in 2 episodes
Core — 6× in this paper · explained in 2 episodes
Loss Function — 5× in this paper · explained in 2 episodes
Aggregation — 4× in this paper
Calibration — 4× in this paper · explained in 7 episodes
Data Assimilation — 4× in this paper · explained in 2 episodes
Hilbert Space — 4× in this paper · explained in 2 episodes
Transformer — 4× in this paper · explained in 2 episodes
Hyperparameters — 3× in this paper · explained in 2 episodes
AUC — 2× in this paper · explained in 2 episodes
Accuracy — 2× in this paper · explained in 1 episode
Chaotic Dynamics — 2× in this paper · explained in 2 episodes
Computational overhead — 2× in this paper · explained in 2 episodes
Uncertainty Propagation — 2× in this paper · explained in 3 episodes
Accountability — 1× in this paper · explained in 2 episodes
Axioms — 1× in this paper · explained in 2 episodes
Bayesian Inference — 1× in this paper · explained in 5 episodes
Consistency — 1× in this paper · explained in 3 episodes
Continuous Ranked Probability Score (CRPS) — 1× in this paper · explained in 2 episodes
Diffusion Model — 1× in this paper · explained in 10 episodes
Diffusion Models — 1× in this paper · explained in 9 episodes
Generative Model — 1× in this paper · explained in 2 episodes
Generative Models — 1× in this paper · explained in 3 episodes
Objective Function — 1× in this paper · explained in 2 episodes
Parameterization — 1× in this paper · explained in 2 episodes
Representation Learning — 1× in this paper · explained in 3 episodes
Reproducing Kernel Hilbert Space (RKHS) — 1× in this paper · explained in 7 episodes
Robustness — 1× in this paper · explained in 5 episodes
Sparsity — 1× in this paper · explained in 4 episodes
Time Series Forecasting — 1× in this paper · explained in 4 episodes
Transformers — 1× in this paper · explained in 2 episodes
Verification — 1× in this paper · explained in 2 episodes

Summary

The gist A distribution-to-distribution (D2D) neural probabilistic forecasting framework is developed to operate directly on predictive distributions, enabling recursive propagation of uncertainty within a unified end-to-end neural architecture for dynamical systems.

Framework Overview

The core idea of the proposed distribution-todistribution (D2D) framework is to treat predictive distributions, rather than point states, as the fundamental objects of dynamical forecasting <ref:2603.25370#pg2> In this formulation, both inputs and outputs of the forecasting model are predictive distributions, enabling uncertainty to be propagated explicitly and coherently through successive forecast steps <ref:2603.25370#pg2>. The framework consists of three main components: 1) A distribution encoder, which maps input distributions into a finite-dimensional representation suitable for neural processing <ref:2603.25370#pg2>, 2) A replaceable neural forecasting module, which transforms the encoded representation through standard neural architectures (e.g., LSTM [18], Transformer [19], or other backbone models) <ref:2603.25370#pg2>, and 3) A distribution decoder, which maps the neural outputs back to parameterised predictive distributions <ref:2603.25370#pg2>. This modular design allows existing neural forecasting architectures to be incorporated into the D2D pipeline with minimal structural modification <ref:2603.25370#pg2>.

Encoding and Decoding Mechanisms

The framework utilizes specific mechanisms for encoding input distributions and parameterizing output distributions <ref:2603.25370#pg2>.

**)& Encoding Input Distributions for Neural Processing: A key technical challenge in distribution-to-distribution learning is how to represent input distributions in a form suitable for neural network processing <ref:2603.25370#pg2>. In this work, kernel mean embedding (KME) [25], [26] is adopted to encode input distributions <ref:2603.25370#pg2>. Given a positive definite kernel k: X ×X → R defined on the state space X, the kernel mean embedding of a probability distribution P over X is defined as µk(P):= EX∼P [k(X, ·)] <ref:2603.25370#pg2>. The resulting object µk(P) lies in the reproducing kernel Hilbert space (RKHS) associated with the kernel k [2603.253701]. Under characteristic kernels (e.g., the Gaussian RBF kernel used in this work), this embedding uniquely represents the distribution <ref:2603.25370#pg2>. In practice, the embedding is evaluated at a finite set of centres nc j=1, yielding a finite-dimensional representation zP = (µk(P)(c1),..., µk(P)(cnc)⊤, which serves as the input to the neural forecasting module <ref:2603.25370#pg2>. The kernel bandwidth l plays a critical role in controlling the smoothness and sensitivity of the embedding <ref:2603.25370#pg2>. A further crucial property of kernel mean embeddings is their linearity with respect to the underlying distribution: µk (∑ i wiPi) = ∑ i wiµk(Pi) <ref:2603.25370#pg2>. This linearity plays a central role in the proposed framework <ref:2603.25370#pg2>. The MDN decoder outputs a predictive distribution in the form of a Gaussian mixture consisting of M = 5 components for each of the three state variables <ref:2603.25370#pg2>. This parameterisation is used for the predictive marginal distribution of a single target state variable <ref:2603.25370#pg2>. The predictive marginal density is then given by p(y x) = ∑ M i=1 πi(x) N (y µi(x), σ2 i (x)) <ref:2603.25370#pg2>. This flexibility enables MDNs to capture non-Gaussian and multimodal predictive marginal distributions with minimal additional computational overhead relative to modern neural network architectures <ref:2603.25370#pg2>. The encoding is naturally matched to the MDN-based output representation, ensuring that input and output distributions are expressed within the same functional form <ref:2603.25370#pg2>. The kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. This alignment enables seamless iteration of the D2D mapping, allowing predictive marginal distributions to be propagated consistently through successive forecasting steps <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. The kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. The kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>. Overall, kernel mean embedding provides a principled and computationally tractable mechanism for mapping predictive marginal distributions into finite-dimensional neural representations, while preserving compatibility with Gaussian mixture outputs <ref:2603.25370#pg2>. The framework only requires structural compatibility between the chosen mixture distribution family and the kernel function <ref:2603.25370#pg2>.

Improvements for AI systems

One can implement a distribution-to-distribution (D2D) neural probabilistic forecasting framework that operates directly on predictive distributions rather than relying on ensembles or sampling, allowing for recursive propagation of predictive uncertainty within a unified end-to-end neural architecture.
The improved system can learn a distributional forecasting operator that produces skillful predictive marginal distributions consistent with the underlying system dynamics, effectively making uncertainty an intrinsic component of the model evolution rather than an auxiliary quantity.
The framework enables the training and optimization to be driven explicitly by probabilistic performance metrics, such as minimizing the empirical logarithmic score, which is defined as L = −1/N ∑ N i=1 log p(x obs i)/ for a collection of forecasts.
The system can capture nontrivial distributional evolution under nonlinear dynamics by treating input distributions via kernel mean embeddings and outputting predictive distributions using mixture density networks (MDNs), enabling the production of flexible, non-Gaussian multimodal predictive marginal distributions.
The iterative strategy allows the model to learn a dynamically consistent distributional evolution operator, which is superior to the direct strategy because it avoids applying models to distributions that lie outside the regime on which it was trained, leading to more robust performance across different lead times and noise levels.

Abstract

Probabilistic forecasts provide a principled representation of forecast uncertainty through predictive distributions. Directly propagating predictive distributions with conventional physical models requires solving distributional evolution equations, a task well-known to be computationally prohibitive for high-dimensional nonlinear systems. Consequently, multi-step predictive distributions are commonly approximated using ensembles or sampled trajectories. A novel distribution-to-distribution (D2D) neural probabilistic forecasting framework is proposed with compatible distributional encoding and decoding interfaces around a replaceable neural forecasting module. Kernel mean embeddings represent input distributions, while mixture density networks parameterise outputs, enabling recursive multi-step forecasting in distributional form. The framework is demonstrated on the Lorenz63 system. Results show that the D2D model captures nontrivial distributional evolution, produces skillful probabilistic forecasts without explicit ensemble simulation, and remains competitive with a naively initialised perfect-model benchmark. This provides proof-of-concept evidence that predictive distributions can be recursively propagated through a neural forecasting model and evaluated in terms of probabilistic forecast performance.