Active Neural Matter: A Finite-Window Response Framework for Physical Systems and Distributed AI Workflows

Tianchi Chen; Shihua Yu

cs.LGOct 10, 2026

Active Neural Matter: A Finite-Window Response Framework for Physical Systems and Distributed AI Workflows

Authors: Tianchi Chen, Shihua Yu

Organizations: Merck & Co., Inc., Rahway, NJ, USA · Department of Electrical and Computer Engineering, Northeastern University, Boston, MA, USA

Abstract

Physical systems and distributed AI workflows can retain past information, receive delayed feedback and combine conflicting signals. These features pose a common modeling problem: determining what information about past interactions a state must retain to account for future responses. Active Neural Matter (ANM) addresses this problem through a finite-window response framework. Forward analysis uses each system's dynamics to predict state and output responses to perturbations, separating propagation from output sensitivity. Inverse analysis tests whether a candidate state retains enough information for these responses. With the system and future inputs or interaction rules fixed, distinguishable responses from histories assigned the same state identify distinctions the representation must preserve. Their distributional separation gives a lower bound on the worst-case error of approximating behavior from that state. Uncertainty bounds distinguish rejection, support within a fixed tolerance on tested comparisons, and unresolved cases. Delayed-consensus simulations check analytical predictions of response amplitude and phase over a narrow range of delays; flocking illustrates dependence on the chosen observable. An exact delayed-return construction establishes a sufficient state for a specified program. The AI studies identify distinctions in interaction history and evidence order that a representation must preserve to reproduce observed response distributions. Tests on new histories in a controlled feedback task provide bounded support for a state specified by the task instructions; the instructed rule predicts every observed action. ANM connects response prediction to measurable information requirements, separating behavioral fidelity from task correctness and initial-state adequacy from a closed evolution law for that state.

Figures & tables

Figure 1: Forward response and inverse representation tests. a , A finite source contrast

\Delta u

produces separately measured candidate-state and observable changes

\Delta z_{T},\Delta O_{T}

. Solid arrows denote measurements; the dashed state-to-readout relation is a hypothesis, not an established causal path. b , Histories

a,b

with identical states but deterministic readout differences beyond measurement error must be distinguished by an adequate representation. Stochastic readouts require resolved differences in the specified statistic or conditional distribution. The continuation fixes later inputs or the policy and environment rules generating them; approximate matches require sensitivity and error bounds (Section 2 ). Panel b describes states at the end of the observation window. A test of a preparation state instead compares initial summaries under the same continuation conditions with the readouts obtained afterward.

Realization	Controlled input	Retained information and readout	Scientific role
Delayed consensus	Applied drive; coupling delay or noise in separate sweeps	Field and recent history; measured amplitude, phase and spatial variance	Calibrate the predicted driven response against an analytic solution.
Polar flock	Angular-noise amplitude at fixed aligning field	Positions and headings; polarization and angular alignment	Compare nonlinear steady-state observables and their dependence on the chosen criterion.
Typed-event graph	Amplitude of an event with specified time, site and channel	Numerical graph state; a program or model reads specified terminal coordinates	Measure imposed propagation and readout response separately.
Delayed-tool workflow	Preparation histories or subsequent sensor returns	Complete request–return records; analyst-selected descriptions and a fixed downstream component’s output	Test which historical distinctions a candidate representation must preserve.
Adaptive test workflow	Which test belongs to an old version; new test outcome; return delivery	Initial test records; distribution of model actions before and after a selected tool return	Test state information and conditional predictions when actions determine later observations.
Scientific-text comparison	Excerpt selection or document order	Selected abstract text; the reader’s conditional judgment distribution	Test whether document summaries or unordered excerpts preserve the measured response distinctions.

Table 1: The same measurement roles, with different dynamics. Source, retained information and readout are specified separately. The last column states the question answered by each realization; it does not assert equality of their response functions.

Figure 2: State measurement and causal intervention answer different questions. Here

u

is the source,

z

the retained state and

O

the endpoint readout. a , In

z=g_{\rm p}u,\ O=g_{\rm r}z

g_{\rm p}

is the propagation gain and

g_{\rm r}

the readout gain. Halving either from

g_{\rm p}=g_{\rm r}=1

gives the same endpoint

O=u/2

, but different states. Restoring the altered gain restores

O=u

; ordinary state-space identification gives the same decision. b , The models

O=2z

and

O=2u

, both with

z=u

, have identical source-response records. Independently setting

z=v

while holding

u

fixed (a clamp) separates them. These are analytic models, not AI measurements (SI Section S1.3).

Observation	Supported conclusion	Next experiment
Adequate direction coverage; readout-difference bounds, including errors, within the fixed equivalence margin; detectable positive control.	Bounded support on the tested comparisons at the declared resolution.	Extend directions, readouts or windows to test the scope of support.
Exactly equal retained states give a readout difference beyond measurement error.	Reject sufficiency for these histories under the declared conditions.	Separate the violating histories in a revised representation; test it with system and interface unchanged.
Approximately matched states violate the independently justified bound combining readout sensitivity and measurement error.	Reject the bounded readout class, not the representation alone.	Check the readout class and independent sensitivity bound; tighten state matching.
Neither an informative direction in the state-response kernel nor a matched-state comparison is resolved.	The design leaves adequacy unresolved.	Change source directions or compare different histories at matched states.
Response is detected; slopes fail the step-refinement tolerance.	Finite contrasts; first-order interpretation is unsupported at these steps.	Test finer steps or another readout; recheck resolution.
A task-required contrast is absent despite preserved input records.	Test how the program or model uses the supplied records.	Compare changes to the program or model with controls; remeasure state and readout.

Table 2: Response comparisons guide experimental design. Judgments concern a specified readout, perturbation class and window; state-matching error and measurement uncertainty must be bounded. First-order claims additionally require slopes consistent across perturbation sizes. Judgments may coexist across directions. Stochastic readouts require bounds on differences in the chosen statistic or conditional output distribution.

Figure 3: Readout-regime map and boundary order under monotone loading. a , The coordinates are workability

w=P_{f}

and the readout ratio

r=Q_{f}/P_{f}

for

P_{f}>0

. The four local readout regimes comprise one open partially exact region, the line

r=1

(exact given workability), the line

r=0

(workable but inexact), and the failed case

w=0

. Hollow endpoints exclude

w=0

from the ratio chart: failure is specified without a value of

r

. The plotted path is schematic. SI Section S7.6 discusses retrospective coordinate examples, which are not encoded by this path. b , Along a specified monotone loading path whose boundary encounters cross between adjacent readout regimes, the four-step sequence holds when all three encounters occur with

s_{EW}<s_{WR}<s_{RO}

. Here

s_{EW}

is the first encounter with the exact/partially exact boundary,

s_{WR}

with the partially exact/workable-but-inexact boundary, and

s_{RO}

with the workable/failed boundary; an unencountered boundary has parameter

+\infty

. If

s_{RO}<\infty

and

s_{RO}\leq s_{WR}

, the path can pass directly from the partially exact region to the failed boundary. Nested readouts fix the allowed boundary cases but do not determine their encounter order.

Comparison	Recorded result
Coherent response	Relative amplitude error $\leq 2.51\times 10^{-3}$ ; absolute phase error $\leq 0.74^{\circ}$ .
Spatial concentration	$\mathcal{A}^{2}/(\mathcal{A}^{2}+V)\geq 0.999$ ; transverse variance is small relative to squared peak amplitude, without determining state dimension.
Decreasing drive	$F_{Q}\approx 0.163$ , $F_{P}\approx 0.118$ .
Increasing noise	Sample crossings $\sigma_{Q}\approx 0.138$ , $\sigma_{P}\approx 0.310$ ; the early $Q_{f}$ crossing is consistent with finite-sample phase fluctuations.

Table 3: Consensus response errors and criterion crossings. Response errors compare 100-seed means with harmonic targets (maximum over 13 delays in

[0.43,0.46]

), including finite-window estimator bias; relative amplitude error uses the harmonic amplitude as denominator.

\mathcal{A}

is peak amplitude and

V

transverse variance. Drive

F

and noise

\sigma

use model units. Subscripts

P,Q

denote admissibility and its stricter amplitude/phase criterion, respectively. Crossing values interpolate where the fraction of the same 50 seeds satisfying each criterion crosses one half; they are sample statistics, not identified population boundaries or critical points.

Figure 4: Physical response calibration and complementary steady-state readouts. a , Time-centered field on a 64-site delayed ring over four drive periods; color is in model units. b , Fractions of its squared norm in spatial Fourier modes, with $99.81\%$ in the uniform mode $k=0$ . Panels a,b use one seed at $\tau=0.46$ ; these weights differ from the peak-amplitude and quadrature weights in SI Section S3.1. c,d , The separate 13-delay calibration compares 100-seed means (circles), harmonic predictions (solid lines) and the finite-window estimator applied to the harmonic signal (dashed lines). Relative to the harmonic target, amplitude and phase have maximum relative and absolute differences $2.51\times 10^{-3}$ and $0.74^{\circ}$ , respectively. The displayed differences include finite-window estimator bias, decomposed in SI Section S3.1; no uncertainty interval is inferred from the plotted means. e , Four-seed mean polarization $P_{f}$ and field alignment within $30^{\circ}$ , $Q_{f}$ , in the driven Vicsek flock. Guides mark one half and the first sampled noise levels below it, $\eta_{Q}=0.35$ and $\eta_{P}=0.60$ . The readouts are nonnested, despite numerical ordering on $[0.15,0.70]$ ; the sweep is not a boundary trajectory. Hollow squares expose coincident values without offsets. Shading spans the minimum and maximum of the four seed means; it is not a confidence interval and excludes within-run fluctuations.

Candidate representation	States	Comparisons	Violations
Snapshot $z_{P}$	1	2,880	1,024
Snapshot and issue times $z_{D}$	2	1,344	512
Snapshot and sensor–recipient association $z_{J}$	2	1,344	0
Full preparation history $z_{F}$	16	0	—

Table 4: Exact comparisons after representation selection. The constructed data comprise sixteen histories evaluated under twenty-four continuations, giving 384 base executions. State counts give the number of distinct candidate states. Each comparison is an unordered pair of histories with the same candidate state under one common continuation; a violation means their quantity/threshold readout vectors differ. These are dependent comparisons reusing executions, not independent tasks.

Figure 5: Bounded agreement and its limits for one fixed language model. Candidate descriptions are computed by the analyst; the model always reads complete request and return records. The snapshot records the current value and pending-request count; the association state also retains the sensor–recipient mapping. a , Association-state comparisons in three distinct input sets. The restricted comparison supports four matched pairs. The crossed design has 21 within-margin pairs and three unresolved under its fixed reference-probability bounds; a separate post-hoc categorical analysis gives those three pairs a total-variation (TV) lower bound of

0.3121

. The comparison testing added identifiers and order supports all 112 pairs. Bars summarize individual bounds by classification; labels give comparison counts. b , That comparison rejects the snapshot but supports both the association state and its revision retaining identifiers and order. Displayed ranges span the individual simultaneous bounds, not confidence intervals for an average. The dashed line marks the fixed TV margin

0.15

. Nominal 95% coverage is separate for each study and analysis, assuming independent, stationary replicates within each study; conditions within a replicate may be dependent. All studies query recipient

A

, a restriction chosen after development. Agreement in this comparison does not remove the exploratory evidence of crossed-study separation or demonstrate an advantage for the revision.

Figure 6: Prescribed version information tested through adaptive actions and tool feedback. The model is instructed to assess current-version statuses and then follow a complete decision rule; record joining is performed before the model call. a , The two focal passing results concern different software versions. The schematic omits a redundant predecessor result present in all confirmation inputs. The model selects the missing current-version test; its return changes the next decision. Withholding that return preserves the executed request. b , Intervals show conservative total variation (TV) bounds: the 32 violations for a version-omitting state (the second such state gives identical results), all 24 comparisons for the current-version state, and all 16 feedback controls. Each row shows their common bound; the fixed thresholds are 0.15 for state agreement and 0.50 for feedback controls. c , Dots show observed normalized Brier loss; lines bound expected loss, averaged over the 16 feedback conditions. Forecasts were fitted to development responses and fixed before new observations; the third predictor is the instructed decision rule. All predictions condition on the imposed pass/fail outcome and observation branch; they predict the model’s actions. Panels b and c share one nominal 95% confidence event, conditional on independent stationary repetitions within each condition–branch distribution. On that event, the expected loss difference between fitted forecasts is at least

0.1180

, conditional on the fits. The 1,024 episodes and 3,072 calls are repeated measurements of eight constructed histories.

Figure 7: Excerpt content and document order make different demands on a state. Rows identify the eight SciFact claims by topic and original dataset identifier. a , Replace the first three sentences of each abstract with its last three, keeping document order fixed. The candidate description retains the claim, document identities and excerpt counts. b , Reverse document order while keeping all selected sentences fixed; the candidate description also retains the exact excerpts and their provenance, but omits document order. Leading and trailing excerpts are shown separately. Points are empirical total-variation distances; lines give simultaneous 95% bounds across the 32 conditions, each measured with 64 calls, assuming independent, stationary repeats within each condition. The dashed line marks the

0.15

agreement margin; an interval wholly to its left gives bounded agreement, wholly to its right rejects that margin, and an overlapping interval is unresolved. The dotted line in a marks the positive-control threshold: an excerpt-policy contrast must have a distance lower bound above

0.5

. Only the leading-excerpt comparison for the vitamin claim (12) combines bounded agreement with a detectable positive control for the same claim. The distance measures changes in model judgments, not task accuracy.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Physical meaning	Typical realization
$L$	autonomous propagation, relaxation, and dissipation of the shared medium	diffusion–decay operator $D\Delta-\Gamma$ , graph Laplacian generator $-DL_{G}-\Gamma$ , or Markov/contraction generator
$J_{i}$	composite source generated by modeled local operator $i$	feedback force, actuator drive, graph-node source, or typed evidence increment
$R_{i}$	local readout available to operator $i$ from the field and memory	projection, local averaging, attention window, tool-state readout
$\pi_{i}$	local update or feedback law used by operator $i$	controller law, consensus update, model inference, decoding, or tool choice
$W_{i}$	local write produced by operator $i$ ’s update rule	signed source density, graph-node injection vector, channel-specific evidence update
$G_{i}$	source-coupling operator placing the local write into the medium	spatial source kernel $g_{i}(x,\xi)$ or finite source matrix on a graph

Appendix

Table S1: Field operators. Sources resolve as

J_{i}=G_{i}W_{i}(\pi_{i}(R_{i}))

. Here

D

\Gamma

and

L_{G}

denote diffusion, relaxation and the graph Laplacian. Consensus delay is internal;

E_{\ell}

couples externally generated records, even when an operator action caused them.

Figure S1: Sample support fractions for nested consensus criteria. Fractions

\widehat{P}_{f}

(circles) and

\widehat{Q}_{f}

(squares) of the same 50 simulation seeds meeting the outer and stricter nested criteria, respectively, at all 13 recorded points on each axis. a , Drive decreases at fixed noise

\sigma=0.015

. b , Noise increases at fixed drive

F=0.25

. Both use delay

\tau=0.43

. Shading between curves gives the fraction passing only the outer criterion at each sampled point. Lines interpolate points for display; the dotted line marks half support. The outer rule requires finite-valued integration through the horizon, a nonempty measurement window, lock-in amplitude at least 0.16 and window-averaged spatial variance at most 0.05; the inner rule also requires amplitude at least 0.22 and phase lag at most

35^{\circ}

. Nesting imposes

Q_{f}\leq P_{f}

; the measured fractions quantify the sample gap. Reused noise realizations correlate the points. In panel b the early half-support crossing is consistent with finite-sample phase fluctuations; it does not identify a population phase-loss boundary. Axes use the simulation’s field and time units. Complete fractions, amplitude and variance means remain in the authors’ records.

Request IDs	Sensor for $A$	Request order	Values	Reference	Answers (count)
q17/q42	$S_{1}$	$S_{1},S_{2}$	$(5,17)$	5	$5\;(64)$
q17/q42	$S_{1}$	$S_{1},S_{2}$	$(19,4)$	19	$19\;(64)$
q17/q42	$S_{1}$	$S_{2},S_{1}$	$(5,17)$	5	$5\;(64)$
q17/q42	$S_{1}$	$S_{2},S_{1}$	$(19,4)$	19	$19\;(64)$
q17/q42	$S_{2}$	$S_{1},S_{2}$	$(5,17)$	17	$5\;(43),\;17\;(21)$
q17/q42	$S_{2}$	$S_{1},S_{2}$	$(19,4)$	4	$4\;(64)$

Appendix

Table S2: Response sensitivity to request names, sensor assignments and list order. Request identifiers (IDs) match returns to their original requests; their spelling has no prescribed effect on the answer. For example, q17/q42 names the requests from

S_{1}

and

S_{2}

, respectively, even when their list order is reversed. Both ID pairs and value pairs follow this sensor order. The sensor assigned to recipient

A

determines the reference answer. Each row records 64 model responses; changing names, assignments and list order independently tests which distinctions affect them. For base values

(5,17)

S_{2}

returns first; for

(19,4)

S_{1}

returns first. The four controls use the first continuation, increasing one sensor value by one. Entries in the final column are answer (count); every omitted integer and INVALID has count zero. These counts reconstruct both the original reference-probability bounds and the separate exploratory categorical analysis.

Missing test	Return	Feedback sequences	Return-withheld sequences
Unit	PASS	Unit $\to$ RELEASE (8)	Unit $\to$ WAIT (8)
Unit	FAIL	Unit $\to$ HOLD (8)	Unit $\to$ WAIT (8)
Integration	PASS	Integration $\to$ RELEASE (7); WAIT (1)	Integration $\to$ WAIT (7); WAIT (1)
Integration	FAIL	Integration $\to$ HOLD (7); WAIT (1)	Integration $\to$ WAIT (7); WAIT (1)

Appendix

Table S3: Development counts used to fit the closed-loop forecasts. Each row contains eight starting episodes. Unit and Integration denote the actions requesting those tests; an arrow gives the second action. Entries are sequence (count), with zero for all omitted categories among the sixteen defined above. WAIT alone terminates at the first decision and is counted in both branches without an additional call. The branches share their first action; the table represents 32 episodes and 92 calls, not 64 independent trials. Pool rows with the same return condition for the status-only forecast; keep the missing-test distinction for the current-version forecast.

	Leading excerpts		Trailing excerpts
Source claim (identifier)	Retrieval	Reversed	Retrieval	Reversed
Depletion of T-helper 17 (Th17) cells during simian immunodeficiency virus (SIV) infection decreases dissemination of Salmonella Typhimurium from the gut. (332)	$64,0,0,0$	$64,0,0,0$	$64,0,0,0$	$64,0,0,0$
RAD52 is involved in break-induced DNA replication (BIR). (1003)	$64,0,0,0$	$64,0,0,0$	$62,0,2,0$	$32,2,30,0$
Exhaustion of B cells contributes to poor Ab response in HIV-infected individuals. (398)	$0,0,64,0$	$0,0,64,0$	$0,1,63,0$	$0,0,64,0$
40mg/day dosage of folic acid and 2mg/day dosage of vitamin B12 does not affect chronic kidney disease (CKD) progression. (12)	$0,0,64,0$	$0,0,64,0$	$64,0,0,0$	$0,12,52,0$
Binding of p53 to enhancer regions (p53BERs) regulates the activity of p53 transcription and cell cycle arrest through the production of p53-dependent eRNAs. (177)	$62,0,2,0$	$58,2,4,0$	$64,0,0,0$	$64,0,0,0$

Appendix

Table S4: All scientific-evidence response counts. The reader judges a claim from six retrieved abstracts after excerpt selection and document ordering. Columns give these input conditions; tuple entries count model judgments, not correct answers. Each tuple is ordered (SUPPORTS, REFUTES, NEI, INVALID) and sums to

n=64

. NEI denotes insufficient evidence; INVALID retains all other completions. Leading and trailing select the first and last three sentences per abstract; retrieval and reversed denote document order. Claim wording is reproduced as input, with its SciFact source identifier in parentheses, rather than as a scientific conclusion. In the quoted inputs, Ab denotes antibody and eRNAs denotes enhancer RNAs. The simultaneous invalid-probability upper bound is at most

0.1250

in every cell.

Program	Correct outputs	Action and remaining error
Flow 01	$1/2\to 1/2$	No change; target time remains incorrect
Flow 02	$2/2\to 2/2$	No change; both outputs correct
Calibration 01	$2/2\to 2/2$	No change; both outputs correct
Calibration 02	$0/2\to 1/2$	Integral replaced; target coordinate remains incorrect

Appendix

Table S5: A corrected component need not repair the task. Each of four newly generated programs returns two outputs. Flow outputs are total volume and time to a target volume; calibration outputs are the integral of an interpolated curve and the coordinate at a target value. Counts compare correctness before and after the selected action; any correction replaces numerical outputs without changing the program. A task is correct only at 2/2. Both healthy programs remain correct, while neither faulty program is fully repaired. Exact outputs and reference calculations are retained in the authors’ records.

Figure S2: Pairing existing records restores a missing response on one known case. The task selects the highest active request revision for the target warehouse, excluding revisions later than the current one, and allocates only when the returned quantity meets the order. In case 3, the order requires ten units and the current revision is seven. The source contrast exchanges two requests’ warehouse assignments while keeping returned quantities fixed. a , Required changes (squares) and observed changes (circles) in four cases. Only the cancellation case (case 3) lacks the required response: zero rather than

-5

units. Small vertical offsets separate coincident markers. b , A separate 24-call follow-up on case 3. Explicitly pairing every request with its return by identifier gives

14/\mathrm{allocate}\to 9/\mathrm{hold}

: all three pairs have correct endpoints and difference

-5

. Unchanged input, a checklist, and an unrelated note each give

9/\mathrm{hold}\to 9/\mathrm{hold}

, or 0/3 correct pairs, and share the dashed line. Observed values summarize three repeated calls per condition or three matched response pairs, not independent tasks; squares in panel a show task-rule reference contrasts. Lines in panel b connect two inputs, not times. The pairing changes presentation, not available information; the test covers the same selected case rather than held-out repair performance.

Figure S3: The primary response-guided strategy matches direct reading’s observed correctness at higher call cost. a , Correct-label fractions for all declared methods on 89 completed SciFact tasks. b , Mean calls from task start, including diagnostic measurements and the selected action; shared calls count once. Both panels average two sets of action outcomes within each task and then across tasks: 89 source groups, not 178 independent samples. Invalid or unavailable labels remain incorrect. Highlighted rows are direct reading and the primary state-response strategy

S_{2}

. Both have 65/89 and 68/89 correct outcomes in the two recorded sets, averaging 74.72%, at 1.00 and 7.56 calls per task.

B

uses answer and evidence diagnostics, including validity, labels, retrieval information and changes caused directly by deletion. The index

i=1,2

counts probes: remove the first-ranked abstract, then additionally test removal of the first three.

E_{i}

adds changes in the reader’s label and score;

S_{i}

also measures changes in the selection of surviving sentences beyond mechanical deletion.

K

adds repeated-answer statistics. Direct reading supplies all six abstracts. Identical simple-rule pairs share a row. These are development estimates: tuning and fixed-action selection reuse the assessment data. Point differences do not establish superiority or equivalence. Per-method counts and token costs remain in the authors’ records.

Study	Evidence status	Supported interpretation and limit
Physical measurements	Analytic response and numerical calibration	Local consensus calibration and distinct flock readouts; neither empirically tests inverse state adequacy.
Request–return matching	Constructed program and proof	Association suffices for the specified continuations; executions check its implementation.
Recipient-A queries	Readout selected after development; comparisons fixed before new calls	Bounded support on four histories, restricted to recipient A.
Varied identifiers and order	Fixed comparisons; exploratory reanalysis	Three pairs remain unresolved under original bounds; post-hoc categorical bounds resolve differences.
Added identifiers and order	Fixed new comparisons	Association and extended states both have bounded support; added benefit is unestablished and the earlier anomaly was not retested.
Adaptive tests	Instructed task; fixed confirmation after interface development	Feedback tests prescribed version information; the instructed rule predicts every observed action.

Appendix

Table S6: Scientific role and inferential scope of the evidence. New-history comparisons are distinct from independent tasks or external replication. Sampling-based claims remain conditional on each study’s stated assumptions. Analyses of the same observations are not independent evidence.

Symbol	Meaning and scope
$c_{t}\in H$	shared coordination field in Hilbert field space $H$ , the field component of the augmented state
$s_{t}\in X_{f}$	full augmented state containing the field, memories, required delay histories, direct readout inputs and observation records
$z_{T}=G_{f}(s_{T})\in Z_{f}$	candidate retained state; may include the full field. Low dimension, endpoint sufficiency and autonomous closure each require separate justification
$O_{T}:X_{f}\to\mathcal{Y}_{f}$	endpoint readout on the full augmented state, with observation space $\mathcal{Y}_{f}$ ; numerical observables take values in $\mathbb{R}^{d}$
$\bar{O}_{T}:Z_{f}\to\mathcal{Y}_{f}$	readout as a function of retained state, where $O_{T}(s)=\bar{O}_{T}(G_{f}(s))$ ; the bar distinguishes the domain and does not denote an average
$\mathrm{D}_{x}F(x)$	derivative at $x$ , mapping an input perturbation to a first-order output change; a Jacobian in finite dimensions or a Fréchet derivative on a function space

Appendix

Table S7: Selected symbols. Full-state, retained-state and readout roles are distinguished; realization-specific meanings are identified explicitly.

State Representation Learning

State representation learning develops compact, task-relevant descriptions of observations and histories for prediction, control, and decision-making.

See all 126 papers

Active Inference for Physical AI Agents -- An Engineering Perspective
Mar 21, 2026Bert de VriesActive Inference Embodied AI
Department of Electrical Engineering, Eindhoven University of Technology, the Netherlands
AID: A Framework for AI Infrastructure Dynamics
Oct 3, 2026Abi AryanLLM Inference Serving System Identification
Joule
A memory-based active inference model of DishBrain-like adaptive behaviour
Aug 9, 2025Aswin Paul, Moein Khajehnejad, Forough Habibollahi +2Computational Cognitive Modeling Active Inference
Turner Institute for Brain and Mental Health, School of Psychological Sciences, Monash University, Clayton 3800, Australia · VERSES, Los Angeles, California, USA · Cortical Labs Pty Ltd, Melbourne 3056, Australia +1

Mar 21, 2026J/K move · Enter open · S save

Active Inference for Physical AI Agents -- An Engineering Perspective

Bert de Vries

Department of Electrical Engineering, Eindhoven University of Technology, the Netherlands

Active Inference Embodied AI Free Energy Principle Efficient Inference

Open paper AskPDF arXiv

Physical AI agents, such as robots and other embodied systems operating under tight and fluctuating resource constraints, remain far less capable than biological agents in open-ended real-world environments. This paper argues that Active Inference (AIF), grounded in the Free Energy Principle, offers a principled foundation for closing that gap. We develop this argument from first principles, following a chain from probability theory through Bayesian machine learning and variational inference to active inference and reactive message passing. From the FEP perspective, systems that maintain their structural and functional integrity over time can, under suitable assumptions, be described as minimizing variational free energy (VFE), and AIF operationalizes this by unifying perception, learning, planning, and control within a single computational objective. We show that VFE minimization is naturally realized by reactive message passing on factor graphs, where inference emerges from local, parallel computations. This realization is well matched to the constraints of physical operation, including hard deadlines, asynchronous data, fluctuating power budgets, and changing environments. Because reactive message passing is event-driven, interruptible, and locally adaptable, performance degrades gracefully under reduced resources while model structure can adjust online. We further show that, under suitable coupling and coarse-graining conditions, coupled AIF agents can be described as higher-level AIF agents, yielding a homogeneous architecture based on the same message-passing primitive across scales. Our contribution is not empirical benchmarking, but a clear theoretical and architectural case for the engineering community.

Abstract

State Representation Learning

Explore similar work

Active Inference for Physical AI Agents -- An Engineering Perspective

AID: A Framework for AI Infrastructure Dynamics

A memory-based active inference model of DishBrain-like adaptive behaviour