A probabilistic framework for irreversible kinetics

summary

The gist

The gist A probabilistic framework for irreversible kinetics is proposed in which a constrained path functional J encodes constitutive physics and observations on the admissible history space Had,

In short

A probabilistic framework is proposed to unify forward-in-time evolution and inverse inference for irreversible kinetics. It uses a cost functional J and a Gibbs measure, weighted by epistemic uncertainty Theta, to rank admissible histories as Bayesian posteriors. This approach reveals that enforcing causality in Generalized Standard Materials incurs an additional history-level cost compared to simultaneous minimization.

Key concepts

Constrained Path Functional (J): This functional encodes the physics of the material and any constraints from observations. It quantifies the 'cost' associated with a specific sequence of states or history, representing how physically plausible that path is according to known laws and data.
Gibbs-type Probability Measure: This measure assigns probabilities to candidate histories using an exponential function related to the cost functional J and a temperature parameter Theta. It acts as a Bayesian posterior, where higher probability means the history is more likely given the physics and observations.
Epistemic Uncertainty (Theta): Theta controls how much uncertainty exists over the unknown admissible histories. A high Theta allows for broader exploration of possibilities, while decreasing Theta forces the system to concentrate its probability on histories with lower cost J, effectively narrowing down the possible outcomes.

Terminology used across episodes

Reinforcement Learning (RL) — explained in 25 episodes
Retrieval-Augmented Generation (RAG) — explained in 20 episodes
Sample Complexity — explained in 20 episodes
Flow Matching — explained in 18 episodes
Federated Learning — explained in 17 episodes
Quantum Fisher Information (QFI) — explained in 17 episodes

This episode discusses

The paper

A probabilistic framework for irreversible kinetics · Read on arXiv

Manas V. Upadhyay

Laboratoire de Mécanique des Solides (LMS), École Polytechnique, Institut Polytechnique de Paris, CNRS

A probabilistic framework for irreversible kinetics that unifies forward-in-time evolution and inverse inference, while quantifying epistemic uncertainty over admissible histories, is proposed. The forward-in-time evolution and inverse inference formulations differ only through observations and how they constrain admissible histories. All histories are ranked using a Gibbs-type probability measure (-J/Θ), where J encodes constitutive physics and observations. Θ controls epistemic uncertainty over the unknown histories, and the measure is interpreted as a Bayesian posterior over histories. Maximizing this posterior is equivalent to simultaneous minimization of J over the admissible history space. The framework is then specialized to generalized standard materials (GSM), for which J is constructed from incremental energy--dissipation functionals. This specialization reveals an important difference between the history resulting from simultaneous minimization over entire histories generally differs from the one generated by sequential GSM updates. Specifically, the causal GSM history incurs a higher history-level cost than the one obtained by simultaneous minimization, indicating that enforcing causality carries an additional cost at the history level. This difference is shown through seven representative examples: viscoelasticity, viscoplasticity, rate-independent plasticity, gradient damage, conserved and non-conserved phase transformations, and coupled thermoelectric evolution. Finally, the finite- Θ formulation is demonstrated on an inverse problem with a non-convex free energy, where incomplete initial-state information is used to infer unobserved intermediate states and quantify uncertainty over admissible histories. As Θ decreases, the posterior concentrates towards minimum- J histories.

DOI: 10.1016/j.jmps.2026.106875

Transcript

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

Tom: I'm Tom.

Jane: And I'm Jane, here with Lu, Meng and our in-house large language model, Lalam.

Tom: We're looking at the paper: "A probabilistic framework for irreversible kinetics".

Jane: The gist A probabilistic framework for irreversible kinetics is proposed in which a constrained path functional J encodes constitutive physics and observations on the admissible history space Had,

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

Paper summary: Tom: So this paper proposes this probabilistic framework for irreversible kinetics. The main idea is they have this constrained path functional J that encodes the physics and all the observations on a set of admissible histories, Had.

Jane: It seems like it’s trying to connect forward-in-time evolution with inverse inference, which are usually treated separately in modeling.

Lu: They say that the forward evolution and inverse inference only differ based on how observations constrain those admissible histories.

Meng: The core mechanism is a discrete Gibbs-type measure proportional to exp(-J over Theta).

Tom: And this Theta parameter controls epistemic uncertainty, which they interpret as a Bayesian posterior over histories.

Jane: So maximizing that posterior just turns into minimizing J simultaneously over the entire history space, distinguishing between the continuous minimizer hcont and the discrete maximum a posteriori MAP history hMAP.

Lu: The framework is specialized for generalized standard materials or GSM types. For those, J is constructed from incremental energy dissipation functionals.

Tom: And this specialization really shows a difference between the history you get from minimizing over entire histories and what you get from sequential GSM updates.

Jane: Specifically, they find that the causal GSM history has a higher cost at the history level than the one obtained by simultaneous minimization, which indicates that enforcing causality carries an additional cost there.

Meng: That's interesting for production because it suggests that if we are trying to infer something from data—say, an unknown initial state—we need to account for this extra cost when we try to force a causal path.

Tom: It’s about quantifying uncertainty over those unknown histories using this measure.

Jane: So, the paper is setting up a way to rank different possible paths based on how well they fit what we observe.

Conclusion: Tom: So, looking at the full scope of "A probabilistic framework for irreversible kinetics," it seems like they’ve built a method to rank entire admissible paths and not necessarily select the causal GSM history as its MAP history.

Jane: At finite Theta, this framework gives us posterior statistics over complete histories, which they showed using a 1D Swift-Hohenberg system with a non-convex free energy where you only have partial information about the initial state <ref:2601.10763#pg3>.

Lu: As Theta decreases, the posterior distribution of J contracts toward its low constrained path action edge.

Meng: It seems like this is useful for things where we don't know everything about the starting point but we want to quantify how uncertain we are about all those possible paths.

Tom: The authors show that this framework provides a history-level inference method that doesn't have to select the causal GSM history as its MAP history.

Jane: They also demonstrate it using a Metropolis-Hastings algorithm applied to a Swift-Hohenberg non-convex energy problem where the final state is prescribed and only partial information about the initial state is available.

Lu: They admit that they can extend this framework further, which opens up several avenues for future exploration in this area.

Meng: It’s about getting better at inferring intermediate states when we don't have perfect data upfront.

Tom: The title itself, "A probabilistic framework for irreversible kinetics," points to a method for handling complex materials where the path matters more than just the end result.

Jane: So, this is about moving beyond just finding one deterministic solution and understanding the whole space of possibilities when things are messy or unknown.

Lu: It’s about having a way to rank all those admissible paths and quantifying uncertainty in unobserved initial and intermediate states without making assumptions beyond what the constitutive model already states.

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