MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data

arXiv:2604.00333 · math.NA, cs.LG, cs.NA, physics.comp-ph · Submitted 2026-04-01 · 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: "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data".

Jane: The gist: The proposed MVNN framework introduces a measure-valued neural network to infer measure-dependent interaction (drift) terms directly from particle-trajectory observations in mean-field limits,

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

Title and authors: Tom: So we're diving into this paper today. It’s called "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data". It’s essentially about using a neural network to figure out the interaction part of a system directly from watching particles move.

Jane: That sounds dense, Tom. What is the core idea here? How does this actually work in simple terms?

Tom: Well, the main thing they're doing is taking those particle trajectories and using an architecture that operates on probability measures to infer the drift terms—that’s how fast things move—which depend on the overall distribution of all particles. They call this a measure-valued neural network.

Lu: It’s motivated by thinking about functionals on spaces of probability measures, which is like characterizing things based on the entire cloud of particles at once, using finitely many test functions, according to sixty-nine, seventy.

Meng: So they aren't just looking at two particles bumping into each other; they are learning the big picture interaction term, the drift b theta(x, mu) which depends on both where a single particle is and the whole measure mu of all particles.

Tom: Exactly. And what’s really neat is that this way of defining it scales linearly with the number of particles N, meaning it’s O(N), which is a huge win over traditional methods that are quadratic, O(N two) <ref:2604.00333#pg3>.

Jane: That computational efficiency sounds really important for real-world applications where you can't simulate every single pair of agents. So, how do they train this thing to learn the correct interaction?

Tom: They formulate the learning objective by minimizing a mean-squared regression loss, which is derived from discretizing the path measure and is equivalent to maximizing a log-likelihood function. This loss is then reduced into a regression problem where they try to find the best parameters theta that minimize this discrepancy between what their network predicts and what they actually observed.

Lu: They approximate this loss using Girsanov’s theorem, which basically helps them reduce it to fitting the learned drift theta(X t l,m, t l,m) to the true interaction b(X t l,m, t l,m) over time steps.

Meng: That sounds like a clever way to turn a complex statistical problem into a regression task that standard neural network training can handle. But what about the theory behind it? Is this just some fancy fit?

Tom: No, they establish well-posedness of the resulting dynamics and prove propagation-of-chaos guarantees for the corresponding learned particle system. This connects the microscopic particle level to the macroscopic mean-field model when N goes to infinity, confirming that their learned system behaves like a chaotic one.

Title and authors: Jane: That addresses a big theoretical hurdle—showing that the learning process actually results in a mathematically sound dynamic system, not just a curve that fits the data.

Tom: And they also show universal approximation for these cylindrical neural functionals of the form phi integral x, phi emb(times; theta emb), mu; theta int to approximate any continuous drift function b: R d times P two(R d) to R d with arbitrarily small error <ref:2604.00333#pg3>.

Lu: Under the low-dimensional measure-dependence assumption, Theorem two point seven gives them a constructive finite-dimensional approximation result for that target drift function b, and those rates depend on the Lipschitz constants of g and G.

Meng: So, theoretically, they’re saying this network structure has the potential to learn any kind of continuous mean-field interaction, not just simple pairwise ones. That’s a big generalization.

Tom: It is. Now let's look at how they tested it on real systems, because theory is cool but implementation matters a lot. The paper shows numerical experiments on various dynamics.

Jane: What kind of dynamics were they testing? Were these just simple models, or did they tackle something more realistic?

Tom: They tested deterministic and stochastic Motsch-Tadmor dynamics, which involves clustering and consensus formation, as well as 2D attraction-repulsion aggregation and Cucker-Smale dynamics <ref:2604.00333#pg3>. They even looked at a hierarchical multi-group system with the Multi-Group MVNN extension.

Lu: On the multi-group extension, the MG-MVNN successfully learns asymmetric and hierarchical communication structures among different groups of agents by using group specific embedding networks to summarize population features into moment vectors.

Meng: That’s practical for things like social networks or power grids where connections aren't symmetrical; they can capture that asymmetric coupling encoded in influence matrices.

Tom: They even extended it to second-order systems, where the drift b is a function of position, velocity, and the law of the pair in P two(R d times R d), which means learning how velocity affects movement too <ref:2604.00333#pg3>.

Jane: It sounds like they are tackling very complex physics and social structures. What did you find when you compared this MVNN approach to other methods?

Tom: They compared it with Gaussian process models, and the MVNN showed greater efficiency and improved accuracy in predicting complex dynamics because it can handle a large number of agents N while keeping simulation times roughly constant as N increases.

Lu: They also noted that they can infer the intrinsic interaction operator rather than just overfitting to the geometry of the training data, which is important because it suggests a more fundamental understanding of the underlying physics or behavior.

Tom: So, to wrap up, this MVNN framework gives us a data-driven way to efficiently infer complex mean-field interactions from empirical trajectory data without needing an explicit functional form for those forces. It’s robust across many different physical scenarios tested numerically.

Title and authors: Jane: It seems like they’ve built a very solid mathematical and computational structure here that bridges the gap between microscopic particle movement and macroscopic collective behavior.

Tom: Exactly. The paper, "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data", provides a powerful tool for inferring these complex mean-field interactions directly from trajectory data without needing to assume any specific pairwise interaction kernel.

Lu: It suggests that we can move past restricting our hypothesis class to just two-body forces when modeling emergent collective behaviors in biology or social systems.

Meng: From an engineering standpoint, the O(N) complexity is what makes this approach viable for large simulations where explicit pair interactions are computationally prohibitive.

Lalam: I see a vision here where AI can learn the governing equations of complex physical systems directly from raw observation data, which means we could build models that understand emergent collective phenomena in ways we can't currently program by hand.

Tom: That’s the big picture there. We’ve seen how this MVNN framework works and what it can do for modeling complex dynamics, but it also leaves room for future work.

Jane: What does the authors themselves say about what this method stops short of? Where does it not apply?

Tom: They point out that in many biological, social, or physical scenarios, the effective dynamics are governed by a drift of the mean field form—a nonlinear functional of the population distribution—rather than just a simple superposition of two-body forces.

Lu: They mention that restricting the hypothesis class to pairwise interaction kernels can therefore limit our ability to capture emergent collective behaviors in those real-world settings.

Meng: So, it’s not a magic fix for every complex interaction; it’s powerful when the underlying structure is indeed mean-field based.

Lalam: This paper sets a strong foundation for how AI can learn continuous governing equations directly from particle observations, and I think that ability to learn intrinsic operators is really important for improving cultural understanding in complex systems.

Tom: That’s the essence of it. We’ve covered the MVNN framework, its learning mechanism, its theoretical guarantees, and the impressive empirical results on everything from swarming models to hierarchical groups.

Jane: It's a lot to take in, but it really shows how deep we can get into understanding how large systems evolve based on local interactions.

Tom: Indeed. We’ve just discussed "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data" and what it means for modeling complex dynamics with AI.

The paper's summary: Tom: So we're looking at what they actually said in the summary of this paper, "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data." Essentially, they’re saying this new neural network structure lets us figure out those complicated mean-field interactions—the drift terms—directly from watching particles move around.

Jane: That’s right. Think of it like a system where the movement of one particle depends on the overall shape of the crowd, not just on its immediate neighbors, and this MVNN is how we learn that dependency. It takes those trajectory observations and turns them into a mathematical model for how the entire system should behave as it evolves.

Lu: What I find really interesting is how they handle the math behind it. They use these cylindrical features and an embedding network to get a distribution-to-vector representation, which makes the learning process scalable, meaning it can handle a lot of particles without getting bogged down in massive calculations.

Meng: Scalability is key for us. If we’re simulating big systems, we need something that runs efficiently. And they claim the evaluation of this learned drift scales linearly with N, which is a huge deal compared to the quadratic cost you get from modeling every single pair interaction explicitly.

Tom: That linear scaling is what I want to emphasize because it moves us away from those slow, expensive pairwise models and toward something that can actually handle big-scale simulations. But it’s not just about speed; they also proved some important theoretical stuff on the theory side.

Jane: They showed well-posedness for the resulting dynamics, which means the system they learn actually makes sense mathematically and won't just fly off into nonsense when you simulate it over time steps.

Lu: And they proved propagation-of-chaos guarantees for their learned particle system, so we know that if we start with a small number of particles and let N go to infinity, the dynamics they learn actually behave like a chaotic mean-field model, which is what we expect in many physical systems.

Meng: That connects the microscopic particle level to the macroscopic behavior in a very rigorous way, which is exactly what engineers need when we’re trying to build predictive models for complex phenomena.

Tom: And theoretically, they gave us universal approximation theorems. This means this network structure isn't just good at fitting one specific type of interaction; it can approximate almost any continuous drift function we want, given the right assumptions about how dependent the interaction is on the measure.

Jane: That's powerful because it suggests that if you have observational data, this AI framework has the potential to learn the true governing equation for a wide range of scenarios without having to guess what that equation looks like first.

Lu: They also gave us constructive approximation rates under low-dimensional assumptions, which means we can actually predict *how well* this network will do based on how complex the underlying physics is, using things like Lipschitz constants.

Tom: It’s a solid framework for learning continuous governing equations directly from data without needing to assume a specific two-body interaction kernel upfront.

Jane: So what does this mean for us who are just trying to build simulations or models? It means we can stop guessing the interaction terms and start letting the data tell us what the forces are.

Meng: Practically, it means we can model things like crowd dynamics or swarming behaviors where the rules aren't just simple repulsion between two people, but something more complex that depends on how crowded a whole area is.

Lu: And for social sciences or biology, it opens up the door to modeling those emergent collective behaviors that arise from huge numbers of agents interacting in a way that’s too messy to define with simple equations.

Tom: We've seen they tested this on Motsch-Tadmor dynamics and hierarchical groups, and the results were strong, showing it captures clustering and asymmetric communication structures well.

Jane: So basically, this MVNN framework is a data-driven way to infer complex mean-field interactions from particle trajectories without needing an explicit functional form for those forces. And now we’re looking at how they might take this further by learning higher-order interaction terms or even reduced representations of the two-particle distributions.

Tom: Right, so it's a robust method for learning continuous governing equations directly from discretized particle observations without needing to assume any specific pairwise interaction kernel. Next up, we’ll look at some other interesting papers that are looking at how memory and structure get encoded in AI models.

The paper's improvements: Tom: We’re now looking at what they suggested to make the MVNN even better, focusing on their improvements and what that means for future research. They’re essentially saying this architecture isn't just a static tool; it has room to grow significantly.

Jane: It’s about taking this core idea of inferring drift from trajectories and making it more flexible, less restrictive, and capable of handling more complex physics. They are pushing beyond just pairwise interactions.

Lu: One big improvement is that they extended the framework to second-order systems, which means they can now learn how velocity affects movement too. The drift term b isn't just a function of position x and the measure mu; it’s also a function of velocity v, so it maps to the full dynamics in two dimensions, position and velocity.

Meng: Learning second-order dynamics is tough computationally, but if they can do it, that opens up modeling for things like swarming models where speed itself changes based on density or local flow. That’s a huge practical step.

Tom: And they also addressed the limitation we talked about earlier—the restriction to simple pairwise forces. They suggest that instead of limiting ourselves to those two-body forces, we should focus on learning the actual effective interaction operator itself.

Jane: That means we stop trying to fit a fixed mathematical form for how particles interact and start letting the network discover the underlying physical law governing that interaction directly from what we observe.

Lu: They also have this idea about reducing representations of two-particle distributions, which is a way to make the input to the network less messy while still retaining all the necessary information about pair interactions. That’s a clever way to handle high-dimensional probability distributions without losing accuracy.

Meng: From an engineering standpoint, reducing that representation complexity is great because it lowers the required memory and speeds up training time for these large particle systems we discussed earlier.

Tom: And they mentioned that this whole framework is a weak-form variant of operator learning, which connects it to other architectures like DeepONet or FNO, but they emphasize that what’s special here is learning the intrinsic interaction operator itself rather than just overfitting to the geometry of the training data.

Jane: So, once you get good at this method, you're not just getting a prediction tool; you're getting a way to understand the fundamental rules of how particles influence each other in a system.

Lu: This is what I see for cultural impact—if we can model complex social dynamics with this level of precision, it could help us design systems that are more resilient and responsive to emergent collective behaviors across different groups.

Tom: It’s definitely pushing the boundaries of how AI can learn the continuous governing equations of physical systems from raw data. Next up, we’re looking at some other papers that are looking at how memory and structure get encoded in AI models.

Conclusion: Tom: So we’re wrapping up our look at "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data." Basically, this paper shows a new way to use neural networks to infer the complex, mean-field interaction terms in systems just by watching particles move.

Jane: It really boils down to using trajectory data to build a model of how the whole crowd behaves collectively over time, without needing some pre-set formula for every single pair interaction.

Lu: The main takeaway is that this method can handle a lot of complexity, like hierarchical groups and asymmetric communication structures, which was pretty impressive in their experiments.

Meng: From an engineering standpoint, that linear scaling with N is what makes it feasible for large-scale simulations where you usually have to stop because the computational cost becomes too high.

Tom: Yeah, it’s efficient and theoretically sound on multiple fronts—it establishes well-posedness and propagation-of-chaos guarantees for the learned system.

Jane: It shows that we can bridge that gap between tiny particle movements and the big picture mean-field dynamics in a way that feels mathematically rigorous.

Lu: And the universal approximation theorem means this architecture is flexible enough to learn almost any continuous drift function, given certain assumptions on how dependent it is on the distribution of particles.

Tom: It’s a powerful data-driven method for learning continuous governing equations directly from observations without needing an explicit functional form for those forces.

Jane: So what does this mean for us who are just trying to build simulations or models? It means we can stop guessing the interaction terms and start letting the data tell us what the forces actually are.

Tom: We've seen they tested this on Motsch-Tadmor dynamics and hierarchical groups, and the results were strong, showing it captures clustering and consensus formation well.

Jane: It really is a strong framework for learning continuous governing equations directly from discretized particle observations without needing to assume any specific pairwise interaction kernel.

Lu: Future work could involve extending this to learn higher-order interaction terms or even reduced representations of the two-particle distributions to model beyond mean-field dynamics.

Meng: I'd like to see more work on how they can apply this for real-time, adaptive simulations where the system parameters change as it evolves.

Tom: That’s a good direction for future research, looking at dynamic systems that aren't static.

Jane: This paper is a great example of how AI can learn the intrinsic operator rather than just overfitting to the geometry of the training data.

Lu: I think this work points toward a future where AI can truly model emergent collective phenomena in ways we can't currently program by hand.

Meng: It's certainly pushing how we use machine learning for physics and complex systems modeling.

Tom: We’ve just discussed "MVNN: A Measure-Valued Neural Network for Learning McKean-Vlasov Dynamics from Particle Data," a method that shows how AI can learn the governing equations of physical systems directly from raw observation data.

Jane: And next up, we’re going to look at some other papers that are looking at how memory and structure get encoded in AI models.

Liyao Lyu, Xinyue Yu, tracy Schaeffer

University of California, Los Angeles

math.NA, cs.LG, cs.NA, physics.comp-ph

Submitted: 2026-04-01

Updated: 2026-10-09

License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/

Importance score: 80/100

The gist: The gist: The proposed MVNN framework introduces a measure-valued neural network to infer measure-dependent interaction (drift) terms directly from particle-trajectory observations in mean-field

Key concepts

Measure-Valued Neural Network (MVNN): This architecture extends conventional neural networks to work with probability measures instead of simple vectors. It uses an embedding network to create scalable distribution-to-vector representations, allowing the network to learn functions that depend on the entire population's distribution, not just individual points.
Cylindrical Features: These are the specific mathematical features learned by the MVNN architecture. They allow the network to capture complex dependencies on probability measures by integrating test functions against these features. This enables the network to represent functionals defined on spaces of probability measures.
Propagation-of-Chaos: This is a theoretical guarantee proving that as the number of particles ($N$) grows, the learned mean-field model converges to the true interacting particle system. It confirms that even though we are using a simplified mean-field approximation, the underlying chaotic nature of the microscopic system is preserved.
Mean-Field Limit: This refers to the situation where a large number of interacting particles ($N o ext{infinity}$) are considered. In this limit, the complex interactions between every pair of particles simplify into a predictable, continuous drift term that depends only on the overall distribution (the measure) rather than individual pairwise forces.

Terminology

Aggregation — 1× in this paper
Generalization — 1× in this paper · explained in 13 episodes
Vector Representations — 1× in this paper · explained in 2 episodes

Summary

The gist: The proposed MVNN framework introduces a measure-valued neural network to infer measure-dependent interaction (drift) terms directly from particle-trajectory observations in mean-field limits, providing a data-driven alternative to traditional kernel estimation methods.

How it works

The MVNN architecture extends conventional neural networks to operate on probability measures by learning cylindrical features, using an embedding network that produces scalable distribution-tovector representations The architecture is motivated by the cylindrical functional framework [69, 70] which characterizes functionals on spaces of probability measures via finitely many test-function integrals The MVNN drift is defined as bθ(x, µ):= φintx, φemb(·; θemb), µ; θint, where µ ∈ P2(R d), and the interaction network maps the local state and the global (measure) feature to the drift Evaluating this drift scales linearly with N, i.e., O(N), which contrasts with explicit pairwise interaction models that incur a quadratic computational cost of order O(N 2)
The learning objective is formulated by minimizing the mean-squared regression loss (11) derived from the discretization of the path measure, which is equivalent to maximizing the log-likelihood function (8) This loss function is approximated via Girsanov’s theorem and then reduced to a regression problem: θ⋆ ∈ arg min θ 1/MLN X M m=1 L X−1 l=0 X N i=1 bˆθ(Xi tl,m, µˆtl,m) − b(Xi tl,m, µˆtl,m) 2. (13)

Theoretical Guarantees

(a)

The framework establishes well-posedness of the neural network induced McKean-Vlasov dynamics and proves propagation-of-chaos guarantees for the corresponding learned particle system This bridges the gap between microscopic and macroscopic descriptions by showing that the N-particle system (4) converges to the mean-field model (5) as N → ∞, confirming that fθ N t is ft-chaotic

(b)

A universal approximation theorem shows that cylindrical neural functionals of the form (2) can approximate any continuous drift b⋆: R d × P2(R d) → R d with arbitrarily small error Under the low-dimensional measure-dependence assumption, Theorem 2.7 provides a constructive finite-dimensional approximation result for the target drift function b⋆(X, µ), establishing approximation rates dependent on the Lipschitz constants of g and G

Empirical Validation

(a)

Numerical experiments validate the accuracy of the learned model on various systems, including deterministic/stochastic Motsch-Tadmor dynamics, 2D attraction-repulsion aggregation, Cucker-Smale dynamics, and a hierarchical multi-group system The MVNN successfully captures the clustering and consensus formation inherent in the Motsch-Tadmor model without explicit knowledge of the microscopic interaction form

(b)

The multi-group extension, MG-MVNN, successfully recovers asymmetric and hierarchical communication structures among heterogeneous groups of agents by parametrizing each drift bk using a composite neural operator that maps local state and aggregated population features to the target drift This architecture correctly learns the asymmetric causal structure encoded in the influence matrix in a hierarchical setting

(c)

The framework can be extended to second-order systems, where the drift b is a function of position (X), velocity (V), and the law of the pair (fθ) in P2(R d × R d) The numerical results show that for second-order attraction-repulsion swarming models, the learned model faithfully reproduces the true dynamics, capturing effective potentials that support metastable geometric patterns

(d)

Comparisons with Gaussian process models demonstrate that the MVNN achieves greater efficiency and improved accuracy in predicting complex dynamics, as it can handle a large number of agents N while maintaining simulation times approximately constant as N increases

The MVNN framework provides a data-driven method to efficiently infer complex mean-field interactions from empirical trajectory data without requiring explicit functional forms This approach offers a rigorous framework for learning continuous governing equations directly from discretized particle-level observations The method's ability to generalize well to unseen initial configurations confirms its robustness in physical and biological contexts This work can be viewed as a weak-form variant of operator learning, conceptually related to but distinct from existing architectures like DeepONet or FNO Future work may involve extending the framework to learn higher-order interaction terms or reduced representations of two-particle distributions to model beyond–mean-field dynamics The paper concludes that the MVNN framework successfully learns the intrinsic interaction operator rather than merely overfitting to the geometry of the training data

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MVNN

insufficient for complex, realistic systems. In many biological, social, and physical scenarios, the effective dynamics are governed by a drift of the mean field form, a nonlinear functional of the population distribution rather than a superposition of two-body forces Restricting the hypothesis class to pairwise interaction kernels can therefore limit the ability to capture emergent collective behaviors

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MVNN: A MEASURE-VALUED NEURAL NETWORK FOR LEARNING MCKEAN-VLASOV DYNAMICS FROM PARTICLE DATA Liyao Lyu∗ Department of Mathematics University of California, Los Angeles Los Angeles, CA 90024, USA lyuliyao@math.ucla.edu Xinyue Yu∗ Department of Mathematics University of California, Los Angeles Los Angeles, CA 90024, USA tracy@math.ucla.edu Hayden Schaeffer Department of Mathematics University of California, Los Angeles Los Angeles, CA 90024, USA hayden@math.ucla.edu ABSTRACT Collective behaviors that emerge from interactions are fundamental to numerous biological systems To learn such interacting forces from observations, we introduce a measure-valued neural network that infers measure-dependent interaction (drift) terms directly from particle-trajectory observations The proposed architecture generalizes standard neural networks to operate on probability measures by learning cylindrical features, using an embedding network that produces scalable distribution-tovector representations On the theory side, we establish well-posedness of the resulting dynamics and prove propagation-of-chaos for the associated interacting-particle system We further show universal approximation and quantitative approximation rates under a low-dimensional measure-dependence assumption Numerical experiments on first and second order systems, including deterministic and stochastic Motsch-Tadmor dynamics, two-dimensional attraction-repulsion aggregation, CuckerSmale dynamics, and a hierarchical multi-group system, demonstrate accurate prediction and strong out-of-distribution generalization

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MVNN insuufficient for complex, realistic systems. In many biological, social, and physical scenarios, the effective dynamics are governed by a drift of the mean field form, a nonlinear functional of the population distribution rather than a superposition of two-body forces Popular examples include crowd dynamics governed by local density constraints rather than individual repulsion [12], vehicular traffic where speed is a nonlinear functional of the flow density [13], and cell migration influenced by chemical concentration fields [14]

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MVNN

insufficient for complex, realistic systems.

Improvements for AI systems

Bold header: Data-Driven Inference of Mean-Field Drift from Trajectories

This improvement allows AI systems to infer measure-dependent interaction terms directly from particle observations, moving beyond assuming a binary-interaction ansatz or pairwise interaction kernels which are insufficient for complex systems. The system can accurately predict the drift term in McKean–Vlasov SDEs using the proposed MVNN architecture, achieving accurate prediction and strong out-of-distribution generalization.

Bold header: Scalable and Computationally Efficient Modeling

The MVNN architecture is designed to operate on probability measures by learning cylindrical features and utilizing an embedding network that produces scalable distribution-to-vector representations, resulting in a computational complexity of O(N) for evaluating the learned drift, which is vastly superior to the O(N2) cost of explicit pairwise interaction models.

Bold header: Robust Modeling of Complex Collective Behaviors

The framework enables AI systems to accurately model diverse collective dynamics, including deterministic and stochastic Motsch-Tadmor dynamics, 2D attraction-repulsion aggregation, and hierarchical multi-group system. This allows the AI to capture emergent phenomena like clustering and consensus formation even when trained on different initial configurations.

Bold header: Inference of Asymmetric Multi-Group Coupling

The Multi-Group Measure-Valued Neural Network (MG-MVNN) can infer complex inter-group coupling by using group-specific embedding networks to summarize population features into moment vectors, allowing the model to capture asymmetric and hierarchical communication structures encoded in influence matrices.

Bold header: Extension to Second-Order Dynamics

The framework is extended to second-order McKean–Vlasov stochastic differential equations involving position and velocity, enabling AI systems to learn drift terms as a function of both state and velocity, leading to the approximation of the full dynamics via equation (12).