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Neural Fields Encode Adaptation Geometry

Prateik Sinha, Stefania Druga

arXiv:2610.07253v1cs.LG

TL;DR

Neural fields are often assessed by reconstruction, but that alone does not show how readily a fitted network adapts or what its weights retain. The paper studies local tangent geometry and sequential fitting, finding links between adaptation costs, retained history, and forgetting timescales.

  • Problem

    Reconstruction quality alone does not reveal how a fitted network responds to further updates or whether its weights retain information about how they were reached.

    Source: §1 · p. 1

  • Method

    The paper compares class-specific nonlinear image adaptation with local tangent-kernel predictions and studies history in weights produced by repeated sequential fitting.

    Source: §1 · p. 1, §5 · p. 9

  • Results

    A local model predicts relative adaptation costs, while tangent-kernel eigenvalues link easy-to-change directions with faster overwriting; controlled loops also leave changed geometry at the same sampled outputs.

    Source: §1 · p. 1, §3.3 · p. 5, §4.4 · p. 9

  • Takeaways & Limitations

    Neural-field representations contain information beyond reconstruction in their response to change and in the history retained by repeated fitting.

    Source: §7 · p. 9

    The practical encoder uses nonlinear Adam, and the paper does not establish instantaneous tangent modes as the causal carrier of its full classification advantage.

    Source: §6 · p. 9

Abstract

from arXiv · show

Neural fields are usually evaluated by how well they reconstruct an observation. We show that this misses two useful properties of a fitted network: how easily it can adapt to new observations, and what its weights retain from earlier ones. We study these properties as adaptation geometry. For images, we meta-learn class-specific initializations, adapt each one to a new image, and measure how much the network must change to fit it. A simple local linear model closely predicts this adaptation cost, while replacing one network's tangent kernel with another's substantially worsens the prediction. Adaptation thus depends on the local geometry of the fitted network, not only on its current reconstruction. For physical fields, we repeatedly fit the same network to observations from a sequence. Its weights then retain information about that history. When two wave histories end at exactly the same observation, the final weights recover the sign of the wave velocity with 68.6% accuracy, whereas the current observation alone contains no such information and gives 50%. These two phenomena are quantitatively linked: tangent-kernel eigenvalues predict both which changes are easy to learn and how quickly they are overwritten by later fitting. Together, these results show that neural fields contain useful information beyond what they currently reconstruct: in how they can change and in how they got there.

1 INTRODUCTION

The introduction argues that reconstruction alone does not reveal how a neural field adapts or what its weights retain from earlier observations. It studies both properties and connects adaptation geometry to memory.

Local geometry predicts adaptation

  • A class-specific local linear model predicts relative nonlinear adaptation costs, and swapping tangent kernels between classes worsens those predictions.The comparison controls for the kernel’s overall scale.Source: §1 · p. 1

Sequential fitting creates useful memory

  • Sequential fitting retains information about earlier observations, demonstrated with wave histories ending at identical observations and physical-regime classification.The paper frames this as useful history in the weights, beyond the current observation.Source: §1 · p. 2

Adaptation geometry and memory

  • The tangent spectrum links adaptation and memory by determining which changes are easy to make and, locally, which are quickly overwritten.Source: §1 · p. 2

Controlled geometry beyond reconstruction

  • A controlled closed loop can return a network to the same sampled reconstruction while leaving its parameters and tangent kernel changed.This establishes that sampled outputs do not uniquely determine local adaptation geometry.Source: §1 · p. 2, Figure 1 · p. 2

2 RELATED WORK

Related work covers neural-field priors and local sensitivity, sequential representations, and memory in partially observed dynamics. This paper examines whether local tangent geometry predicts adaptation and whether sequential fitting retains useful history.

Neural fields, learned initializations, and local geometry

  • Neural fields encode signals with coordinate networks, while learned initializations, fitted weights, and local sensitivities have been used as representations or for adaptation (Nichol et al., 2018).The paper tests whether prior-specific tangent kernels predict rankings and choices of finite-budget nonlinear adaptation.Source: §2 · p. 3

Sequential neural representations and memory

  • Sequential warm-starting of neural fields appears in prior work, including NIRVANA (Maiya et al., 2023), while other work studies continual-field forgetting and time-evolving neural representations.The stated contribution is analyzing what reconstruction-based fitting leaves in weights and whether that history is useful.Source: §2 · p. 3

Memory in partially observed dynamics

  • The Mori–Zwanzig formalism describes reduced dynamics with a memory integral and unresolved-state term, motivating memory for partially observed learned PDE surrogates (Buitrago Ruiz et al., 2025).The paper connects this perspective to optimization history by testing whether sequential fitting provides memory-bearing weights and whether adaptation geometry also governs retention.Source: §2 · p. 3

3 METHODS

The methods use learned network initializations, compare local linear adaptation costs with nonlinear fitting, encode physical sequences through repeated fitting, and analyze how local geometry relates to retention and output-space loops.

3.1 MEASURING WHICH OUTPUT CHANGES ARE CHEAP

  • For image experiments, the method learns a prior for each class, adapts it to a new image, and compares nonlinear adaptation energy with a local tangent-kernel approximation.The prior is a learned initialization, trained with Reptile; lower energy means a lower reconstruction-and-movement objective, not necessarily fewer optimizer steps.Source: §3.1 · pp. 3–4, §3 · p. 3
  • The tangent kernel defines accessible output directions, and its eigenvalues weight the linearized adaptation cost so that directions with larger eigenvalues are cheaper to change.This is an exact minimum for the linearized objective using the finite network, not an infinite-width approximation.Source: §3.1 · p. 4

3.2 ENCODING A SEQUENCE BY CONTINUING THE PREVIOUS FIT

  • For physical sequences, sequential fitting starts each frame from the previous frame’s weights; independent fitting restarts from a shared prior, with matching architecture and fitting budget.The optimizer state resets each frame, so information passes through weights rather than retained Adam momentum.Source: §3.2 · p. 4
  • The physical-regime readout compares raw-frame, independent-weight, and sequential-weight sequences using standardized, PCA-compressed features and class-specific ridge-regularized linear next-step models.The same classification procedure is applied across representations, with PCA rank and ridge strength selected on validation data.Source: §3.2 · p. 4

3.3 LOCAL GEOMETRY SETS MEMORY TIMESCALES

  • Under frozen-Jacobian gradient descent, a tangent-mode perturbation retains a fraction of its size after s steps, with larger-eigenvalue modes decaying faster.This controlled linearized result does not describe the practical nonlinear Adam encoder directly.Source: §3.3 · pp. 4–5
  • The closed-loop test follows target fields using minimum-norm parameter velocities and checks whether parameters and the tangent kernel return when the final target matches the first.Numerical checks and controls are provided in Appendix G.Source: §3.3 · p. 5

4 EXPERIMENTS AND RESULTS

These experiments test what fitted neural fields encode beyond reconstruction: their local capacity to adapt, the history retained by sequential fitting, and the geometry connecting adaptation and forgetting. Across image and physical-field tasks, local geometry predicts adaptation, sequential weights can reveal past dynamics, and controlled experiments examine retention and whether identical sampled outputs determine that geometry.

4.1 LOCAL GEOMETRY PREDICTS NONLINEAR ADAPTATION

  • On MNIST-100, the M-layer’s own kernel predicts full-adaptation rankings at 0.955 correlation and prior choice at 88.3% agreement, versus 0.886 and 75.7% for shared geometry.Reassigning another class’s kernel reduces correlation to 0.770 for the M-layer and 0.466 for SIREN; the gap persists after equalizing kernel traces.Source: §4.1 · p. 5, Table 1 · p. 6

4.2 DIFFERENT HISTORIES REMAIN DISTINGUISHABLE AT THE SAME OBSERVATION

  • With identical final wave displacements but opposite velocities, sequential endpoint weights yield 68.6% mean sign accuracy, versus 50% by construction for observation-only and independent-fit controls.The sequential-weight gain over independent weights is positive in all nine settings; the decoded field and endpoint residual reach 55.6% and 56.4%.Source: §4.2 · p. 6, Figure 2 · p. 7

4.3 SEQUENTIAL WEIGHTS IMPROVE PHYSICAL-REGIME CLASSIFICATION

  • Sequential weights outperform raw fields and independent weights on Gray–Scott and shear flow, while explicit raw delays match the sequential result on Allen–Cahn.Gray–Scott reaches 64.9% versus 59.3% for validation-selected raw delays; shear flow reaches 34.8% versus 27.7%, while Allen–Cahn gives 64.4% versus 65.4%.Source: §4.3 · pp. 7–8, Table 2 · p. 7

4.4 THE SAME GEOMETRY CONTROLS ADAPTATION AND FORGETTING

  • Matched tangent updates yield 46.5% versus 30.1% mean accuracy on Gray–Scott, but matched and mismatched updates are near chance on shear flow.Removing the penalty for moving from the preceding fit does not remove the gain, while other tested controls do not improve performance; the Jacobian diagnostic is system-dependent.Source: §4.4 · p. 8, §4.3 · p. 8
  • The tangent-kernel spectrum governs both local reachability and forgetting: high-eigenvalue directions are easier to change but are overwritten faster than low-eigenvalue directions.The local retention fraction for a mode with curvature κ_i after s steps is ρ_i(s) = (1 − ηκ_i)^s.Source: §4.4 · p. 8
  • Predicted and measured mode-wise retention have median rank correlations of 0.986 on Allen–Cahn and 0.945 on Gray–Scott, but slow modes are not shown to carry useful classification memory.Output-matched selective erasure does not consistently hurt performance more when slow modes are removed; differentiating through actual Adam fits becomes unstable over an eight-frame horizon.Source: §4.4 · p. 8
  • More fitting steps monotonically improve reconstruction, while regime-classification accuracy peaks at an intermediate budget.This budget trade-off is reported for regime tasks; the wave experiment shows no washout over its tested 25–500 steps per frame.Source: §4.4 · p. 8, Table 3 · p. 9
  • A controlled closed loop returns sampled outputs to within median relative error 3 × 10^-14 while changing parameters and the tangent kernel.This existence result uses a complete 12 × 12 grid at the two smallest loop scales; ordinary SGD and Adam do not reproduce the same transport signatures.Source: §4.4 · p. 9

5 DISCUSSION

The discussion argues that fitted neural fields should be evaluated beyond reconstruction. It connects local adaptation and retained history through the tangent spectrum, and notes that identical sampled outputs need not determine adaptation geometry.

5 DISCUSSION

  • The tangent spectrum links two properties beyond reconstruction: how a fitted network responds to updates and what repeated fitting retains.The controlled-loop experiment separately shows that fixing sampled outputs does not, in general, fix local adaptation geometry.Source: §5 · p. 9

6 LIMITATIONS

The paper links tangent geometry to memory timescales but leaves open how fully it explains memory from practical sequential fitting. Its scope is also constrained by the readout, endpoint representation, and dependence of geometry on parameterization.

6 LIMITATIONS

  • The practical memory mechanism remains unresolved: the retention rule is exact for frozen linearization under controlled gradient descent, while the encoder uses nonlinear Adam updates.Selective erasure does not identify slow instantaneous modes as the useful-memory carrier, and the finite-horizon test failed numerical prerequisites.Source: §6 · p. 9
  • The sequence experiments establish usefulness under one matched class-conditioned dynamics readout, not superiority to every classifier on raw sequences.In the endpoint-collision test, terminal observations match exactly, but sequentially decoded fields are not perfectly identical.Source: §6 · p. 9
  • The geometry depends jointly on the network, loss, and parameter-movement metric, and its generality beyond the tested settings remains unknown.Broader architectures and modalities are needed to assess how generally the geometry–memory relationship holds.Source: §6 · p. 9

7 CONCLUSION

The paper argues that reconstruction alone misses a fitted neural field’s capacity to change and its retained history. Its appendices specify the adaptation objective, local linear analysis, kernel controls, and coordinate dependence.

7 CONCLUSION

  • A fitted network has a measurable local response to change, and repeated fitting can leave useful history in its weights; reconstruction quality alone is not a complete representation evaluation.The conclusion recommends evaluating these properties rather than assuming better reconstruction necessarily gives a better representation.Source: §7 · p. 9

A.1 THE OBJECTIVE AND ITS IMPLEMENTED NORMALIZATION

  • The adaptation objective combines reconstruction error with a penalty for moving from a learned initialization, with a positive-definite matrix setting direction-dependent movement costs.For image experiments, the implemented objective uses an isotropic movement metric; the finite-step score is the lowest encountered objective value, not a guaranteed global minimum.Source: §A.1 · p. 13

A.2 SOLVING THE LINEARIZED ADAPTATION PROBLEM

  • The linearized adaptation problem replaces the network response near its initialization with a Jacobian-based model and expresses the cost in output coordinates.Residual directions aligned with large kernel eigenvalues cost less than equally large residuals along weak directions; the prior output centers the equal-cost ellipsoids.Source: §A.2 · p. 13

A.3 RELATION TO THE NONLINEAR OBJECTIVE NEAR THE PRIOR OUTPUT

  • The tangent energy is the quadratic Taylor term of nonlinear adaptation energy near the prior output, but this local result does not ensure accurate predictions for large adaptations or finite-step optimization.The calculation uses the trained finite network and does not assume infinite width or linearity throughout nonlinear adaptation.Source: §A.3 · p. 13, §A.2 · p. 13

A.4 SHARED, REASSIGNED, AND SCALE-NORMALIZED KERNELS

  • Reassigning kernels while keeping prior outputs fixed tests whether candidate-specific geometry predicts adaptation, and the advantage persists when kernel traces are equalized.The intervention changes the scoring rule rather than constructing a network with the reassigned kernel; equal traces retain eigenvalue ratios and eigendirections.Source: §A.4 · p. 14

A.5 DEPENDENCE ON PARAMETER COORDINATES

  • Adaptation geometry is defined by the network, loss, and movement metric together, rather than by the network function alone.A coordinate change requires a corresponding metric transformation to preserve the movement cost; fixing a Euclidean penalty in different coordinates can define a different adaptation problem.Source: §A.5 · p. 14

B.1 DATA SPLITS AND HYPERPARAMETER SELECTION

The appendices detail image-experiment settings and prediction measures, then report kernel comparisons and sequential-fitting controls across image and physical-field tasks. The results concern prediction of adaptation or history under specified readouts, not general classification superiority.

B.4 REPLICATION AND THE DISTINCTION FROM CLASSIFICATION ACCURACY

  • For SIREN, using each prior’s own kernel improves correlation and decision agreement over the shared kernel in all four settings, with gains of 0.255, 0.120, 0.192, and 0.104 in correlation and 18.9, 25.3, 37.7, and 36.0 percentage points in decision agreement.Paired query and fold intervals exclude zero for every setting.Source: §B.4 · p. 15
  • The correct mean–kernel pairing beats reassignment across all eight settings, but own-kernel geometry does not improve every measure over shared geometry.On M-layer Fashion-MNIST, own-kernel correlation is 0.906 versus 0.918 for shared geometry, while decision agreement is 89.3% versus 77.0%.Source: §B.4 · pp. 15–16

B.2 CHOICE OF INR ARCHITECTURES

  • After equalizing kernel traces, none of 100 MNIST-30 M-layer derangements matches the correct pairing on accuracy, correlation, or decision agreement; each measure has p = 0.0099.The image experiments use SIREN and M-layer architectures, and prediction metrics measure agreement with nonlinear adaptation rather than ground-truth classification accuracy.Source: §B.2 · p. 15, §B.3 · p. 15, §B.4 · p. 16

B.5 INDEPENDENT-REFIT KERNEL TRANSFER

  • Independent-refit transfers rank same-class donor kernels above different-class donors, while the recipient’s own kernel remains strongest, including after trace normalization.This pattern holds for within-query rank correlation and decision agreement; own-minus-same-class gaps exclude zero.Source: §B.5 · p. 16

B.6 BOUNDARY CHECKS FOR THE TANGENT SCORE

  • On MNIST-30, tangent scores preserve useful relative score structure despite not accurately reproducing final adapted outputs.Across query–class subsets, mean boundary-check values range from 0.66 to 0.88 and medians from 0.59 to 0.87.Source: §B.6 · pp. 16–17

D.2 MATCHED READOUT AND REPORTING CONVENTIONS

  • Gray–Scott uses two held-out reporting protocols: the primary protocol gives raw fields at 56.3/57.8 and sequential weights at 64.9/65.6, while Table 2 reports raw, independent, and sequential representations at 57.0/53.3, 59.0/55.6, and 64.9/65.6.Values are seed mean / score ensemble; the protocols are kept separate because the sequential result is unchanged while the raw comparator differs.Source: §D.2 · p. 18

D.3 CONTROLS ON WHAT MAKES THE SEQUENTIAL WEIGHTS USEFUL

  • Sequential fitting improves shear-flow reconstruction MSE from about 0.758 to 0.182, but applying the same readout to decoded fields gives 26.8%, versus 44.6% for weights.Removing the movement penalty slightly raises the sequential ensemble to 45.5%; concatenating intermediate checkpoints lowers accuracy to 39.3%.Source: §D.3 · p. 18
  • Explicit raw-frame delays close the Allen–Cahn gap under the tested readout but do not reproduce the Gray–Scott or shear-flow sequential-weight results.Selected delay accuracy is 65.4% versus 64.4% for sequential weights on Allen–Cahn, and 59.3% versus 64.9% on Gray–Scott; on shear flow, selected H = 1 gives 27.7% versus 44.6% for sequential weights.Source: §D.3 · pp. 18–19

D.4 REPLICATION, CLASSWISE BEHAVIOR, AND FORECASTING

Sequential weights outperform raw representations in every tested seed cell, though Gray–Scott performance varies sharply by class. A paired-wave experiment uses identical terminal observations and opposite velocities to test whether fitted weights retain history.

D.4 REPLICATION, CLASSWISE BEHAVIOR, AND FORECASTING

  • The sequential-minus-raw accuracy difference is positive in every tested seed cell: all three shear-flow encoders and all nine cells each for Allen–Cahn and Gray–Scott.Improvements range from 5.0 to 38.7 points on Allen–Cahn, with a minimum of 5.6 points on Gray–Scott.Source: §D.4 · p. 19
  • Gray–Scott sequential accuracy is uneven by class: maze and spots reach 100.0%, worms 97.0%, spirals 85.9%, gliders 6.7%, and bubbles 0.0%.Seed standard deviations are at most 6.4, so the aggregate gain coexists with substantial classwise failure modes.Source: §D.4 · p. 19
  • Gray–Scott held-out one-step R2 stays near zero or negative across ranks: raw peaks near −0.0004, while independent and sequential weights peak near +0.0014.The classification gain does not rely on an accurate low-dimensional one-step forecast.Source: §D.4 · p. 19

E.1 CONSTRUCTION, EXACT BOUND, AND ENCODER AUDIT

  • The paired wave trajectories end with exactly equal displacement fields and opposite velocities, so deterministic terminal-observation features yield an exact 50% sign-classification bound.The bound does not apply to sequential encoders, whose earlier observations differ; the experiment audits data regeneration, wave-equation residuals, equality, and split integrity.Source: §E.1 · p. 19

E.2 ENDPOINT PROBES AND RECONSTRUCTION LEAKAGE

Sequential endpoint weights make wave-history information accessible to a sign probe, though sequential reconstructions are not exactly equal. Longer histories improve endpoint accuracy at 16 frames, but the reported gains are not monotonic with history length or fitting budget.

E.2 ENDPOINT PROBES AND RECONSTRUCTION LEAKAGE

  • 68.6% mean sign accuracy from sequential weights exceeds the 50.0% current-observation and independent-weight controls; velocity regression gives R2 = 0.22 versus zero for current-observation controls.The sign-accuracy gain is positive in all nine settings, with paired bootstrap 95% interval [16.5, 20.6] percentage points.Source: §E.2 · p. 20, Table 9 · p. 20
  • Sequential reconstructions are not exactly equal: decoded-field sign probes reach 55.6% and terminal-residual probes 56.4%, so above-chance weights do not establish exclusively output-invisible memory.Median pairwise reconstruction root mean squared error is 0.00645.Source: §E.2 · p. 20

E.3 FOLLOW-UP: FITTING BUDGET AND HISTORY LENGTH

  • Endpoint sign accuracy is 50.0%, 60.7%, 55.7%, 57.7%, and 68.6% for 1, 2, 4, 8, and 16 fitted observations; the 16-frame condition exceeds every shorter history.The result supports accessible endpoint history but not monotonic accumulation with history length.Source: §E.3 · p. 20
  • From 25 to 500 Adam steps per frame, endpoint-weight accuracy stays between 67.7 and 68.6 while velocity R2 stays between 0.215 and 0.222; this range shows no useful-memory washout.Decoded-field and residual probes follow different budget trends, so the weight signal is not explained simply by measured reconstruction leakage.Source: §E.3 · p. 20

F.1 MATCHED AND MISMATCHED LOCAL UPDATES

Matched tangent updates improve classification on Gray–Scott but do not do so on historical shear flow, despite greater local fidelity there. The section also tests retention predictions, practical fitting budgets, erasure, finite-horizon derivatives, and controlled output-preserving transport, with scope and numerical limits on each.

F.1 MATCHED AND MISMATCHED LOCAL UPDATES

  • On Gray–Scott, matched updates improve classification by 16.4 seed-mean points; on historical shear, matched accuracy is 26.8%, below 28.6% mismatched and 38.4% nonlinear.The diagnostic supports a system-dependent role for tangent identity, not a general causal explanation of nonlinear memory.Source: §F.1 · p. 21

F.2 THE RETENTION RULE AND ITS CONDITIONS

  • Under 0 ≤ ηκ_i ≤ 1, larger tangent eigenvalues predict faster mode-wise decay, and controlled-update retention correlates strongly with predictions on Allen–Cahn and Gray–Scott.Across seed combinations, median correlations range from 0.982 to 0.989 for Allen–Cahn and 0.910 to 0.964 for Gray–Scott; this does not make the formula exact for practical Adam fitting.Source: §F.2 · p. 21

F.3 FITTING BUDGET IN THE PRACTICAL ENCODER

  • Allen–Cahn accuracy peaks at 69.4% with 200 steps then falls to 60.1% at 1,000; Gray–Scott peaks at 69.2% with 50 then falls to 64.1%.Sequential fits lead independent fits at every tested Allen–Cahn budget from five steps onward and at every tested Gray–Scott budget.Source: §F.3 · p. 22

F.4 SELECTIVE ERASURE DOES NOT IDENTIFY SLOW MODES AS THE MEMORY CARRIER

  • Slow-minus-fast erasure damages accuracy by 1.04 points on average, failing to identify the frozen slow quartile as the useful-memory carrier.The decoded-distortion ratio is 1.001; this result does not invalidate the retention rule.Source: §F.4 · p. 22

F.5 DIFFERENTIATING THROUGH MULTIPLE ADAM FITS: NUMERICAL LIMITS

  • At eight frames, replay error rises to 4.69 × 10^-5 and leading-singular-value drift is 0.264, so the intended causal erasure intervention is not performed.The causal question remains unanswered; this is not evidence that finite-horizon memory directions do not exist.Source: §F.5 · p. 22

G.1 EXACT TRANSPORT AND NUMERICAL ELIGIBILITY

  • All nine base-field/initialization combinations satisfy transport prerequisites, including rank 144/144 and QR/SVD parameter-velocity disagreement at most 4.1 × 10^-12; “exact” refers to the sampled-output constraint.The experiment uses the identity parameter metric.Source: §G.1 · p. 23

G.2 FULL LOOP DIAGNOSTICS AND REPRODUCTION

  • Across all 36 loops, median fine-integration relative closure is 1.71 × 10^-12; nonzero parameter and kernel changes and other diagnostics support curvature of the nonlinear parameterization.The tests establish equality of all 144 sampled outputs, not equality of the continuous function away from those coordinates.Source: §G.2 · p. 23

G.3 COMPARISON WITH ORDINARY FITTING

Practical SGD and Adam fitting do not reproduce the characteristic tangent-kernel changes of exact transport in the tested setting. Although practical and exact change magnitudes correlate, their directions and orientation behavior differ.

G.3 COMPARISON WITH ORDINARY FITTING

  • 0.9148 correlation between practical and exact kernel-change norms contrasts with a median directional cosine of 0.0247.Forward/reverse cosines remain positive for SGD (+0.6676), fresh Adam (+0.9071), and carried Adam (+0.9654), unlike exact transport.Source: §G.3 · p. 24
  • Increasing target density from 4 to 32 reduces tracking MSE from 1.6 × 10^-6 to 2.7 × 10^-7, while exact-kernel cosine changes from -0.222 to +0.105.Source: §G.3 · p. 24
  • A zero-loop control yields exactly zero loss, weight change, and kernel change; ordinary fitting therefore fails the exact-transport signatures in this tested setting.The controlled-loop experiment does not show that ordinary SGD/Adam reproduces transport or that geometry edits improve subsequent learning.Source: §G.3 · p. 24

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