Neural Fields Encode Adaptation Geometry

Prateik Sinha; Stefania Druga

cs.LGOct 5, 2026

Neural Fields Encode Adaptation Geometry

Authors: Prateik Sinha, Stefania Druga

Organizations: Machine Learning Department Carnegie Mellon University Pittsburgh, PA 15213, USA · Sakana AI Tokyo, Japan

Abstract

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.

Figures & tables

Figure 1: Adaptation geometry and sequential fitting carry information beyond reconstruction. (a) A class prior is its reconstruction

\mu_{c}

and tangent kernel

K_{c}

that sets which output changes are cheap; reassignment keeps

\mu_{c}

and swaps

K_{c}

. (b) In all settings, the matched kernel predicts nonlinear adaptation far better than a reassigned one (Spearman correlation with nonlinear adaptation energy). (c) Independent encoding restarts every frame from the shared prior

\bar{\theta}

; sequential encoding starts each fit from the previous frame’s weights. (d) With the same class-conditioned readout, sequential weights classify the regime better than raw fields or independent weights on Allen–Cahn, Gray–Scott, and shear-flow (seed means); raw frames with explicit delays match them on Allen–Cahn but not on Gray–Scott. The shear flow delay search selects current frames alone (Table 2 ).

		Spearman $\rho$			Decision agreement (%)
Setting	$\Delta$ Adapt.	Shared	Match	Shuffle	Shared	Match	Shuffle
MNIST-30 M-layer	+12.2	0.829	0.939	0.690	73.3	84.4	61.9
MNIST-100 M-layer	+5.3	0.886	0.955	0.770	75.7	88.3	58.9
Fashion-100 M-layer	+2.0	0.918	0.906	0.791	77.0	89.3	61.9
KMNIST-100 M-layer	+0.0	0.884	0.935	0.771	65.0	79.7	47.7
MNIST-30 SIREN	+4.4	0.618	0.873	0.327	38.9	57.8	11.0

Table 1: The correct local geometry predicts nonlinear adaptation. Correlation measures similarity in score rankings; agreement measures selection of the same prior. Shared uses the mean kernel; Match uses each prior’s own kernel; Shuffle reassigns kernels while fixing initial reconstructions (100 derangements).

\Delta

Adapt. is nonlinear-minus-static classification accuracy in percentage points, a separate measure of utility. Entries are pooled held-out metrics rather than seed means; paired uncertainty for the method differences is reported in Appendix B.4 .

Figure 2: Different histories stay distinguishable at the same observed endpoint. (a) Paired wave histories differ in their earlier frames but end at exactly the same displacement field with opposite hidden velocities. (b) Velocity-sign accuracy of a linear probe on sequential endpoint weights as a function of how many frames were fitted before the endpoint; the observed field and independently fitted weights sit at chance by construction, and the field decoded from the sequential weights and its residual reach 55.6% and 56.4%. Dots at 16 frames are the nine seed cells; the whisker is the paired-bootstrap 95% interval.

System	Raw fields	Independent weights	Sequential weights	Raw + delays	Seq. minus best
Allen–Cahn	42.1 / 48.8	31.9 / 37.5	64.4 / 66.2	65.4	$-1.0$
Gray–Scott	57.0 / 53.3	59.0 / 55.6	64.9 / 65.6	59.3	$+5.7$
Shear flow (45 support/class)	27.7	28.6 / 33.0	34.8 / 44.6	27.7 ( $H=1$ )	$+6.2$

Table 2: Sequential fitting makes weight sequences more useful for classifying dynamics. Accuracy (%) as seed mean / score ensemble; an ensemble standardizes each query’s class-score vector within a run and averages across runs before deciding. Raw + delays selects among stacks of up to 16 raw frames in total: the current frame and at most 15 preceding frames. It reports means over three data seeds for Allen–Cahn and Gray–Scott, and one deterministic run for shear flow, where validation selects

H=1

(no added history). The delay baseline uses the primary evaluation, whose Gray–Scott raw comparator is 56.3 rather than the diagnostic 57.0 shown here. The final column is the sequential seed mean minus the best other mean or deterministic result; differences are computed before rounding. Chance is 25% for Allen–Cahn and shear flow and 16.7% for Gray–Scott. Allen–Cahn and Gray–Scott cross three data seeds with three encoder seeds; shear flow uses one support split and three encoder seeds.

System	Steps/frame	Seq. MSE	Seq. accuracy
Allen–Cahn	50	0.059	64.6
	200	0.043	69.4
	1000	0.035	60.1
Gray–Scott	50	0.679	69.2
	100	0.599	68.8
	1000	0.395	64.1

Table 3: More fitting improves reconstruction but eventually hurts the history-bearing representation. Validation results for sequential fitting. Reconstruction error decreases monotonically with the number of fitting steps, while regime-classification accuracy peaks at an intermediate budget.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Component	Five-field M-layer	Intensity-only SIREN
Outputs	$5\times 28\times 28=3920$	$28\times 28=784$
Adapted parameters	913	2,241
Prior initialization	Common within each fold	Common within each fold
Reptile inner steps	8 Adam steps	8 steps
Reptile inner learning rate	0.04	$5\times 10^{-4}$
Reptile outer steps	1,000	1,000

Appendix

Table 4: Architecture and fitting settings for the image experiments.

Setting	Replication unit	Comparisons	$\Delta\rho$	95% CI	$\Delta$ Agree	95% CI
MNIST-30 M-layer	Query bootstrap	900 pairs	+0.110	[0.093, 0.130]	+11.1	[3.3, 18.9]
	Fold bootstrap	900 pairs	+0.110	[0.090, 0.132]	+11.1	[2.2, 18.9]
MNIST-100 M-layer	Query bootstrap	3,000 pairs	+0.069	[0.061, 0.078]	+12.7	[8.7, 16.7]
	Fold bootstrap	3,000 pairs	+0.069	[0.060, 0.078]	+12.7	[8.3, 17.0]
MNIST-100 M-layer	27 prior sets	2,700 pairs	+0.069	[0.048, 0.095]	+13.7	[5.6, 23.7]
MNIST-100 SIREN	27 prior sets	2,700 pairs	+0.134	[0.096, 0.192]	+33.0	[22.6, 43.3]

Appendix

Table 5: Replication of the own-kernel advantage over shared geometry.

\Delta\rho

is the correlation difference;

\Delta

Agree is the decision-agreement difference in percentage points. Intervals are 95% confidence intervals. The last two rows refit the priors.

	Spearman $\rho$		Decision agreement
Setting	Match	Shuffle	Match	Shuffle
MNIST-30 M-layer	0.943	0.698	82.2	62.6
MNIST-100 M-layer	0.957	0.770	89.0	58.5
Fashion-100 M-layer	0.909	0.797	87.3	62.3
KMNIST-100 M-layer	0.935	0.778	79.0	48.5
MNIST-30 SIREN	0.841	0.362	64.4	10.9

Appendix

Table 6: The reassignment effect persists after equalizing total kernel scale. Each kernel is rescaled to a common trace. Shuffle averages 100 derangements; decision agreement is in percent.

Architecture	Own	Same-class donor	Different-class donor	Shared
M-layer	0.9117	0.8539	0.5549	0.7385
SIREN	0.7847	0.7142	$-0.0802$	0.0917

Appendix

Table 9

Representation	Test (ensemble)	Encoder seeds	Macro-F1	Stored params
Raw states / selected delay ( $H=1$ )	27.7	n/a	0.233	4,224
Independent INR weights	33.0	33.9 / 23.2 / 28.6	0.235	4,612
Sequential endpoints ( $\beta=10^{-3}$ )	44.6	33.0 / 38.4 / 33.0	0.439	676
Sequential endpoints ( $\beta=0$ )	45.5	33.9 / 38.4 / 33.9	0.439	676
Full adaptation path (6 checkpoints)	39.3	33.0 / 37.5 / 31.2	0.384	2,116
Decoded sequential fields	26.8	23.2 / 24.1 / 31.2	0.225	n/a

Appendix

Table 7: Shear flow results and representation controls. Accuracy (%) is shown for the score ensemble and individual encoder seeds. Raw states and the validation-selected delay baseline are the same deterministic representation (

H=1

). Stored parameter counts include the dynamics readout for one encoder seed, excluding PCA arrays.

Representation	Mean	Worst seed	Seed SD	Range	Ensemble
Raw fields	56.3	54.4	1.39	3.3	57.8
Sequential weights	64.9	62.2	1.18	3.3	65.6

Appendix

Table 8: Gray–Scott held-out stability. Accuracy (%) and variation across encoder/data seeds for the primary held-out evaluation.

Data seed	Enc. seed	Sequential	Seq. decoded	Residual	Velocity $R^{2}$	Pair RMSE
14011	101	67.2	56.6	58.2	0.229	0.0062
14011	211	68.0	54.7	50.8	0.216	0.0066
14011	307	68.0	52.0	58.6	0.217	0.0065
25031	101	64.8	57.4	59.0	0.234	0.0064
25031	211	67.6	57.0	54.7	0.222	0.0066
25031	307	68.4	52.7	53.5	0.200	0.0065

Appendix

Table 9: Wave-memory results for each seed cell. Sign accuracy is in percent. Current-field and independent-weight probes are exactly 50% in every row. Velocity

R^{2}

uses sequential weights; Pair RMSE measures the difference between paired sequential reconstructions.

Representation / update	Seed mean	Ensemble	Seed SD
Raw-state dynamics	57.0	53.3	3.8
Independent nonlinear INR	59.0	55.6	4.0
Sequential nonlinear INR	64.9	65.6	1.2
Matched tangent update	46.5	54.4	4.1
Mismatched tangent update	30.1	42.2	6.3

Appendix

Table 10: Replacing the Jacobian changes a local update model on Gray–Scott. Accuracy (%) over nine seed cells, using 128 Jacobian points per transition. Matched beats mismatched in all nine cells. This diagnostic run reports its own raw comparator.

System	Instances	Median Spearman	Median log slope	Slow ret. $s{=}16$	Fast ret. $s{=}16$
Allen–Cahn	216	0.986	1.005	0.998	0.259
Gray–Scott	324	0.945	0.992	0.999	0.579

Appendix

Table 11: Predicted and measured retention under controlled updates. Correlation compares mode-wise retention for each transition. A log-retention slope of one is exact agreement. Slow and fast columns give retention after 16 steps.

System	Steps/frame	Seq. MSE	Indep. acc.	Seq. acc.
Allen–Cahn	1	0.695	26.7	25.7
Allen–Cahn	50	0.059	37.2	64.6
Allen–Cahn	200	0.043	42.7	69.4
Allen–Cahn	1000	0.035	45.1	60.1
Gray–Scott	1	1.128	52.2	56.6
Gray–Scott	50	0.679	55.5	69.2

Appendix

Table 12: More fitting improves reconstruction but not always classification. Validation results. Raw accuracy is fixed at 41.7% for Allen–Cahn and 61.1% for Gray–Scott.

Diagnostic	Ideal	Result
Median complete-field relative closure (fine RK4)	$0$	$2.99\times 10^{-14}$
Closure reduction, coarse $\to$ fine integration	large	$16.5\times$
Net parameter-change local slope in loop scale	$2$	$2.023$
Tangent-kernel-change local slope	$2$	$1.979$
Forward/reverse parameter cosine	$-1$	$-0.999954$
Direct Lie-bracket cosine	$+1$	$0.9948$

Appendix

Table 13: Full controlled-loop diagnostics. Local medians and slopes use nine base-field/initialization combinations at the two smallest loop scales, 0.005 and 0.01 (18 combination/scale records); figure-eight suppression uses the available control records at these scales. The minimum measured tangent-kernel change covers all nine combinations and all four scales.

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Kernel Reboot: Breaking the Boundaries of Neural Tangent Kernels for Neural Fields

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Neural fields (NFs) map continuous coordinates to signals such as color or density, but fast high-quality reconstruction from sparse observations remains difficult. Classical Neural Tangent Kernel (NTK) regression gives closed-form fits, yet it is fundamentally linear and cannot accumulate reusable task priors. We develop three algorithms that address these gaps. NTK-KIP learns a distilled support set of coordinates (and optional labels) so that a finite NTK can inpaint large missing regions from little observed data, yielding a compact non-linear representation instead of a raw kernel solve. MetaQuill meta-learns a shared initialization for an INR so that new scenes can be adapted by updating only a small task-specific weight offset, which provides true feature learning and a reusable prior. Finally, MetaQuill-KIP fuses both ideas: it seeds the task with a KIP-style non-linear warm start, then refines only that small offset around the meta-learned initialization. MetaQuill-KIP achieves high-PSNR reconstructions and semantically plausible inpainting under very sparse observations, while requiring only lightweight per-instance adaptation, whereas diffusion-style baselines typically depend on large pretrained generative priors and costly per-image tuning. This shows that NTK-driven neural fields can be made both non-linear and meta-learnable, narrowing the gap between analytic kernels and practical few-shot reconstruction.

Abstract

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