Toward a General Physics of Learning

We study the lawful relationships between persistent change in a learner and the information, interaction, energy, time, memory, computation, and structure required to produce it.

Physics earns its name through more than borrowed vocabulary. It identifies state variables, makes transformations explicit, discovers constraints that survive across systems, and risks predictions that experiments can falsify.

A general physics of learning should be held to the same standard.

We do not yet have that theory. This document defines the phenomenon we want to study, the distinctions a theory must preserve, and the first experiments that could make the program succeed or fail. The aim is not only to reinterpret existing machine learning. It is to discover principles that explain when finite experience can produce useful behavior in new conditions, then turn those principles into algorithms for artificial, biological, and embodied learners.

TL;DR — We argue a general physics of learning should study lawful relations between persistent learning and its resources — information, interaction, computation, energy, time, memory, and risk — not just rename loss as energy. We define generalization operationally as useful change on a declared set of future conditions, separate four things often called “physics of AI,” define a learning event (s, o, a, b, z, m, U, c), use animated synthetic figures to distinguish geometry, optimization, collapse, and sampling, and propose five falsifiable programs with a 9-part standard for what counts as a law.

Four different projects called “physics of AI”

Several valuable fields are often compressed into one phrase:

1. Physics of hardware: energy, heat, communication, fabrication, and the physical cost of computation.
2. Physics-inspired mathematics: energy-based models, diffusion, statistical mechanics, information theory, dynamical systems, and phase transitions used to analyze learning systems.
3. Physics inside models: known symmetries, conservation laws, simulators, or differential equations incorporated into a model.
4. Physics of learning: general laws and limits governing how experience produces persistent change in a learner.

Our long-term work is the fourth project. The first three supply constraints, mathematics, and experimental systems, but none is by itself a general theory of learning.

The distinction matters. Calling a loss an “energy” does not make optimization thermodynamics. Calling a representation a “manifold” does not make its geometry a property of the world. Useful correspondences must state which quantities correspond, under what assumptions, and what observation would disprove the correspondence.

The phenomenon: a learning event

Let the environment have state $s_t$. A learner receives an observation $o_t$, may take an action $a_t$, and contains a persistent internal state $m_t$. Its update law is

We use learning event for an experience-dependent, persistent change from m_t to m_(t+1) that alters the learner’s distribution of future responses.

The word future is doing the work: those responses are not part of the update itself, so a learning event already concerns interactions the update has not seen.

Improvement on a human-selected score is not required by this definition. An organism can acquire a maladaptive association; a model can absorb a bias; an agent can learn a fact that matters only under a future condition. Performance is an additional measurement placed on learning, not the definition of learning itself.

Persistence is relative to a timescale. A transient activation may be inference at one scale and stored memory at another. A theory must therefore declare the boundary of the learner and the timescale on which change is being measured.

The minimum state vocabulary

These distinctions are not administrative. They prevent a change in belief from being mistaken for a change in the world, a training trajectory from being mistaken for a physical trajectory, and representational proximity from being mistaken for causality.

Why generalization is the central test

A learning event can occur without beneficial generalization. A learner may store one episode, acquire a harmful association, or improve only on conditions it has already encountered. Generalization asks the stronger question: does a persistent change caused by finite experience improve responses in relevant future conditions that were not individually taught?

That question must be scoped. Let $Q$ be a declared distribution over future contexts, interventions, and horizons, and let $\ell(m, q)$ be a lower-is-better response loss for learner state $m$ in condition $q$. For an update from $m_t$ to $m_{t+1}$, define

A positive $\Delta_Q$ means the update improved behavior on that particular set of futures. If $Q$ contains only the training episodes, the quantity measures retention on those episodes. If $Q$ includes new initial states, action sequences, observation channels, tasks, or bodies, it tests a declared form of transfer. There is no context-free generalization score; $Q$, the horizon, the loss, and the learner boundary are part of the claim.

Generalization is central to a physics of learning for four reasons:

1. Finite experience: biological and embodied learners act in worlds whose possible states and trajectories cannot be enumerated one by one.
2. Resource leverage: a useful regularity lets one costly interaction change behavior across many future conditions. The gain must still be compared against the full resource ledger.
3. Structure: transfer is evidence that the update retained an invariance, equivalence, or mechanism shared across the evaluated conditions. It does not by itself reveal which one.
4. Lawfulness: a proposed law of learning must itself survive controlled changes of architecture, task, scale, and substrate. Otherwise it is a local empirical regularity, not yet a general law.

Generalization is not permission to erase differences. Exact memory can be useful, exceptions can be real, and a shift outside the stated scope can invalidate a prediction. The operational target is selective: preserve distinctions that change controlled futures, while avoiding dependence on distinctions that do not matter for $Q$.

What could count as a law?

A physics of learning should search for relationships that are operational across more than one architecture. Some constraints are already well understood in narrower fields; others are research hypotheses.

1. Distinguishability constrains learnability

A learner cannot identify a distinction that never changes the distribution of anything it can observe. If two candidate worlds produce identical observation histories under every admissible action, no learning rule can determine which world it inhabits from those interactions alone.

This is an observability constraint, not an optimizer failure.

2. Computation cannot manufacture external evidence

Additional deterministic computation can extract evidence already present in a history, but without a new informative observation it cannot increase the mutual information between that history and an external hidden variable. Better inference and new evidence are different resources.

3. Learning has a resource vector, not one scalar cost

Two learners can use the same number of transitions while imposing very different physical costs. One may require larger forces, riskier states, more compute, more memory, or longer wall-clock time. Claims of efficiency must specify the target and the complete resource ledger.

A central object of study is therefore not simply “samples,” but a frontier:

The entries in the denominator are not generally interchangeable. Discovering when they can be exchanged, and at what rate, is part of the research program.

Threshold time as a projection of the resource frontier

Time-to-performance is still a useful observable when its scope is explicit. Define the error of learner state $m_t$ on the declared future distribution $Q$ as

and the first passage to an acceptable error $\delta$ as

$C_\delta$ records the resources consumed before the crossing. Sample complexity, wall-clock time, total computation, energy, memory, and intervention risk are different coordinates of this vector. Minimizing observation count is recovered as a legitimate special case when observations have matched costs and the other resources are fixed. Outside that regime, efficient learning is a Pareto frontier over $C_\delta$, not one universal least-time scalar.

4. Representation should preserve controlled futures

A representation may merge two histories only when the distinction it removes is irrelevant to the future behaviors being evaluated. One operational criterion is:

for the relevant action sequences $\mathbf{a}$ and horizon.

This is stronger than visual similarity. It treats representation as an experimentally testable interface: which distinctions must be retained so that intervention and prediction remain correct?

5. Learning operates across coupled timescales

Fast inference, working memory, parameter adaptation, development, and evolution can all alter future responses, but by different mechanisms and at different persistence scales. A general theory should explain their relationships without declaring them identical in advance.

Whether common laws survive across these scales is a hypothesis, not an assumption.

Action as measurement

The first concrete direction in this program is the relationship between action and information.

An action is informative only when it selects an observation channel whose possible outcomes distinguish uncertain alternatives. For uncertain dynamics $\phi$, its one-step epistemic value can be written as

This quantity can be zero. Action does not inherently produce knowledge.

Consider a hidden bit that remains physically fixed. inspect returns the bit; noop returns an independent fair coin. Both actions leave the world state unchanged. inspect yields one expected bit from a uniform prior, while noop yields none. The example separates four events that are often collapsed:

• changing the world;
• changing what is observed;
• changing the learner’s belief;
• changing the learner’s persistent state.

This leads to a falsifiable design principle:

When uncertainty is calibrated and intervention costs are fully charged, actions selected for expected discrimination per unit cost should reduce the interactions required to learn useful dynamics on some declared class of systems.

It is not a universal guarantee. It can fail because the uncertainty model is wrong, the intervention set is uninformative, the hypothesis class omits the real dynamics, or the supposedly informative action creates unacceptable state excursion or risk.

Case study: representation geometry is chosen, not discovered whole

A learned representation can have useful geometry without that geometry being the ontology of the world. A metric says which encoded states are near under a declared criterion. It does not, by itself, say which state causes another, which transitions are reachable, or which structures exist independently of the model.

For a decoder $D$, the pullback metric

measures how local movement in representation space changes decoded output. It is useful for diagnosing sensitivity and interpolation. Causal or physical meaning requires additional evidence: intervention stability, predictive sufficiency, and correspondence with measurements outside the representation itself.

A representation is therefore not automatically a metaphysics. Ontological claims begin when we argue that intervention-stable, invariant structure in a representation corresponds to entities or laws in the world.

Case study: training dynamics is not world dynamics

A scalar loss $L(\theta)$ defines an optimization surface over parameters. Gradient-based training follows a trajectory in parameter space:

This is not the same trajectory as an environment evolving through state space, a belief being revised, or a latent model rolling forward at runtime. Mathematical tools can connect these systems in special cases, but the state spaces and governing laws must be named first.

Backpropagation is a powerful method for adjusting parameters in a differentiable circuit family. Describing it as “circuit search” can be useful. Stronger equivalences require care:

• optimizer path length is not circuit size;
• circuit size is not program description length;
• minimum loss is not automatically maximum probability without a stated model;
• a small solution existing does not guarantee that gradient descent finds it;
• a stationary physical action and a learning objective live in different spaces unless a mapping is explicitly constructed.

The research opportunity is not to declare these quantities equal. It is to discover the conditions under which useful, testable correspondences exist.

A concrete Lagrangian proposal for learning

Guo and Schölkopf propose that learning algorithms are stationary trajectories of a “Learning Lagrangian” in Physics of Learning: A Lagrangian Perspective to Different Learning Paradigms. The paper contains three constructions that should be assigned different evidential status.

First, it minimizes the number of observations required to cross a generalization-error threshold. Under isotropic linear-regression assumptions, this recovers the classical A-optimal design condition $X^\top X=(n/p)I$. This is a scoped result about experimental design and a useful controlled example of action as measurement.

Second, it writes a finite-horizon control problem with costate multipliers and connects the resulting conditions to Bellman optimality. This is an established correspondence between Pontryagin-style optimal control and dynamic programming. It describes return-optimal control, which may occur with known dynamics and no persistent learner update; it is not by itself a law of learning.

Third, it postulates the Fisher-weighted field density

The proposed Euler–Lagrange calculation does not yet derive Adam or RMSprop. Under model expectations, the score and Fisher identities used in the calculation hold at every regular parameter value and select no estimator; under fixed data expectations, those identities do not generally hold together. The action also contains no temporal derivative from which a parameter-update trajectory would follow. The later identification $\dot\theta=F^{-1/2}\nabla_\theta\ell$ is therefore an additional hypothesis, and a diagonal path-dependent Adam update is not generally full Fisher whitening. Likewise, a Legendre transform between reinforcement learning and parameter estimation would require shared conjugate variables and an explicit invertible mapping; none has yet been constructed.

The paper is valuable because it makes a broad intuition concrete enough to test. Recovering known equations is a consistency check, not evidence that all learning follows one physical action.

What a variational mapping must declare

Before comparing a learning problem with mechanics, a variational claim should specify:

1. the configuration variable — data history, environment state, belief, learner state, parameters, or loss field;
2. the path parameter — tokens, updates, interactions, or physical time;
3. admissible paths, endpoint conditions, and allowed variations;
4. the action functional, its units, and its resource interpretation;
5. symmetries and behavior under equivalent reparameterizations;
6. conjugate variables and the transform relating them, if a duality is claimed;
7. a prediction not already implied by generic constrained optimization.

Stationarity alone does not imply a minimum, and the presence of Lagrange multipliers does not make an objective a physical Lagrangian.

Case study: collapse and missing distinctions

Representations can fail by discarding distinctions required for future behavior:

Data distribution influences which distinctions can be learned. Coverage, balance, diversity, and interventions matter because they determine which candidate explanations the evidence can separate—not because a dataset is itself a law of physics. This is the distribution-shift form of the generalization question: which futures beyond the observations can the evidence support, and at what cost?

This reframes the practical question. Instead of asking only how to prevent numerical collapse, we ask:

Which distinctions must the learner preserve, which interactions can reveal them, and what resources are required to retain them persistently?

A program for producing new work

The theory should earn its value by producing experiments and algorithms that would not follow from metaphor alone.

Program I — Information gained per costly interaction

Study active identification under matched resource accounting. Compare passive excitation with actions selected to discriminate among uncertain dynamics. Begin with controlled linear systems where observability and estimation are auditable, then move to partially observed and chaotic systems.

The important output is not “active is better.” It is a map of when the advantage appears, which cost it transfers elsewhere, and which assumptions make it disappear.

Program II — Controlled equivalence and generalization

Develop learning rules that group histories by their action-conditioned futures rather than superficial observation similarity. Evaluate them on a declared $Q$ that varies initial states, interventions, horizons, and observation channels. Test whether the resulting representations improve controlled transfer or reduce interaction complexity after charging all pretraining and adaptation data.

The falsifier is direct: if the representation does not reduce fully accounted learning cost or improve controlled prediction under matched capacity, the proposed principle has not generated an advantage.

Program III — Learning across timescales

Measure when temporary inference becomes persistent memory, when memory becomes a reusable model, and whether common rate limits or invariants survive across in-context adaptation, parameter learning, development, and collective learning.

The first task is not to unify these mechanisms rhetorically. It is to define probes that reveal whether the same candidate relationship holds at each scale.

Program IV — Physical resource frontiers

For embodied learners, count more than transitions. Measure information gained against energy, time, force, state excursion, safety risk, equipment wear, computation, and memory. Search for Pareto frontiers and exchange rates rather than collapsing every cost into one number.

This is where physics of hardware and physics of learning meet experimentally.

Program V — Falsify variational learning laws

Begin where every state and resource is observable. In linear regression, reproduce the A-optimal condition and then introduce unequal observation costs, prefix constraints, model misspecification, and intervention risk. Test whether multi-step planning still improves the fully charged frontier over myopic information gain.

Independently test whether expected surprisal actually decreases across stationary and nonstationary sources, fixed and adapting models, and in-context versus parameter learning. For optimizer claims, compare SGD, Adam, RMSprop, diagonal natural gradient, and full natural gradient under matched data order, total computation, memory, tuning budget, and declared $Q$. Reparameterize equivalent models and ask whether the proposed action and its ranking of trajectories remain invariant. For reinforcement learning, separate planning with known dynamics from online identification in which experience persistently changes the learner.

A variational proposal gains explanatory status only if its typed action predicts an unseen trajectory, failure boundary, scaling relation, or resource tradeoff better than existing optimal-design, control, and optimization accounts. If it merely redescribes an algorithm after the fact, it remains a useful formal interpretation rather than a law.

Our standard for a novel claim

Every proposed principle should specify:

1. System boundary: what is learner and what is environment?
2. State variables: what changes, and on which timescale?
3. Transformation: what update or interaction law is proposed?
4. Resources: what is consumed and how is it charged?
5. Scope: for which systems, future conditions, action constraints, and horizons should the claim hold?
6. Prediction: what observation differs from existing accounts?
7. Falsifier: what result would make us reject or narrow the claim?
8. Coordinate invariance: does it survive an equivalent representation or reparameterization?
9. Dimensional consistency: what are the units, and are combined quantities commensurate?

Without these nine pieces, we have an analogy or research intuition. Those can be generative, but they are not yet a law.

An evidence ladder

Claims should be labeled by the strongest evidence they have earned:

1. Metaphor: shared language suggests a question.
2. Formal correspondence: quantities and assumptions are mapped explicitly.
3. Recovery result: known equations follow from the mapping.
4. Novel prediction: the framework predicts an observation not used to construct it.
5. Cross-system invariance: the prediction survives changes of architecture, task, scale, or substrate.
6. Physical law: operational quantities with units repeatedly survive attempts at falsification within a declared scope.

This separates effective optimizer thermodynamics, such as Neural Thermodynamic Laws, from finite-time ensemble-transport proposals such as A Thermodynamic Theory of Learning I, and from substrate-specific energetic bounds such as Thermodynamic Cost of Inference and Learning in Physical Neural Networks. These works may all be valuable without making the same kind of claim.

What we do not claim

• Latent space is not automatically metaphysics or ontology.
• Backpropagation is not generally shortest-path search.
• The shortest program, smallest circuit, shortest optimizer trajectory, lowest loss, highest probability, least physical action, and lowest energy are not one quantity.
• Runtime computation is not universally a physics engine or path integrator.
• Action does not inherently create information.
• A flat minimum does not universally imply generalization, nor does a sharp minimum universally imply memorization.
• A useful physical analogy is not evidence of a physical law.
• Writing an objective with Lagrange multipliers does not by itself make it a physical Lagrangian, and a stationary path need not be a minimum.

These corrections do not weaken the program. They locate the actual unknowns.

Current uncertainty ledger

Established or source-backed

• Observability limits which hidden distinctions interaction can identify.
• Representation geometry alone does not establish causality or temporal dynamics.
• Deterministic post-processing cannot create new mutual information with an external hidden variable beyond the evidence it processes.
• Environment interactions, computation, energy, time, memory, and risk are distinct resources.

Interpretation

• Controlled predictive equivalence is a useful organizing principle for representations used by acting systems.
• Active experiment design is a natural first domain in which to study learning per costly interaction.
• Variational formulations may unify some acquisition, control, and optimization problems, but the existence and scope of a shared learning action remain open.

Hypotheses

• Some learning constraints remain invariant across artificial, biological, and embodied learners.
• Jointly choosing informative actions and retaining action-relevant distinctions can produce new sample-efficient algorithms.
• Useful exchange laws can be measured between information, interaction, computation, and physical cost on declared system classes.

Unresolved

• What is the right architecture-independent measure of the magnitude of a learning event?
• Which results survive changes in task, learner boundary, and timescale?
• When does an internal representation reflect intervention-stable world structure rather than a convenient coordinate choice?
• Which candidate principles generate an advantage after every source of data and physical cost is charged?

The direction

A general physics of learning is not one grand equation imposed at the beginning. It is a disciplined sequence:

The ambition is broad; each claim must remain narrow enough to fail.

That is what we mean by a general physics of learning: not decorating AI with the language of physics, but building a science precise enough to discover new mechanisms—and demanding enough to tell us when we are wrong.

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Reproducibility note

The figures on this page are synthetic diagnostic animations generated by scripts/generate-learning-figures.mjs; readers who request reduced motion receive the final PNG frame instead. They illustrate distinctions about geometry, optimization, collapse, and sampling; they are not presented as evidence for universal laws. Their checked-in generator keeps explanatory diagrams separate from future empirical results.