cs.AIOct 8, 2026

Prior or Feedback? What an LLM Uses When Adapting Neural Operators

Authors: Julian Chan, Javier Mora Jimenez

Organizations: Compare the Market

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Abstract

Do LLM scientific agents rely only on their initial task context, or do they adapt their decisions in response to experimental feedback? We study this question in neural operator adaptation, where a large language model (LLM) selects fine-tuning configurations under a limited trial budget. Across transfers within and between partial differential equation (PDE) families, the LLM achieves lower held-out test nRMSE than random search and Bayesian optimisation in nearly every matched comparison. Endpoint performance alone cannot distinguish what happens, so we verify each attribution with controlled interventions. Before observing any validation score, the LLM's first configuration already ranks near the top of the corresponding random-search pool, indicating a useful initial bias. A complementary cold-start intervention shows that the selected base learning rate shifts with the PDE description. Once feedback becomes available, reassigning validation scores among evaluated configurations changes the next proposal in every case tested, whereas a value-preserving rewrite produces no comparable aggregate effect. These interventions establish that the LLM's decision-level actions respond to the given task and observed outcomes, showing that it combines a task-dependent prior with sensitivity to experimental feedback.

Figures & tables

Figure 1: Budgeted FNO adaptation and controlled input interventions. (A) Within-family PDEBench shifts used as targets. Lower viscosity ν\nu sharpens Burgers’ gradients, whereas higher advection speed β\beta increases the transport rate. (B) At each of B=20B=20 trials, a policy selects ai∈Aa_{i}\in\mathcal{A} , a fresh copy of the pretrained FNO is fine-tuned, and its normalised root mean squared error (nRMSE) on the autoregressive validation rollout is appended to HiH_{i} . After search, the configuration selected by validation nRMSE is evaluated once on the held-out test set. (C) At cold start, the displayed PDE description is changed; after a history has accumulated, replay interventions reassign validation scores in HiH_{i} . The FNO, data, compute budget, and A\mathcal{A} remain fixed.
FamilySource →\to targetChange
WithinAdvection β=0.1→β∈{0.4,1.0,4.0}\beta{=}0.1\to\beta{\in}\{0.4,1.0,4.0\}speed
Burgers’ ν=1.0→ν∈{0.1,0.01,0.001}\nu{=}1.0\to\nu{\in}\{0.1,0.01,0.001\}viscosity
CrossAdvection β=0.1→\beta{=}0.1\to Burgers’ targetsequation
Burgers’ ν=1.0→\nu{=}1.0\to advection targetsequation
Table 1: Adaptation families. Each source–target cell is evaluated with three seeds.
Target cellLLMRandomTPEIn-domain
Within-family
Adv., β=0.4\beta=0.40.012200.017500.017500.015300.015300.013300.01330
Adv., β=1\beta=10.013400.018100.018100.016600.016600.012800.01280
Adv., β=4\beta=40.007920.015800.015800.009010.009010.008800.00880
Burg., ν=0.001\nu=0.0010.019700.069900.069900.052200.052200.039000.03900
Burg., ν=0.01\nu=0.010.004090.015800.015800.009810.009810.010400.01040
Table 2: Median held-out test nRMSE across three matched seeds after 20 evaluations; lower is better and bold marks the lowest median. †\dagger TPE beats the LLM for seed 43 in this cell ( 0.02630.0263 versus 0.03460.0346 ); the LLM wins the other two seeds. In-domain is a released PDEBench FNO trained on the target regime with about 9,0009{,}000 trajectories, against 750750 per adaptation trial; it is a reference scale, not a matched baseline, and is not bolded.
Figure 2: Controlled input interventions. (a) Distribution of the base learning rate over 9090 cold-start proposals per displayed description, pooling 4545 from each target cell because the same text gives near-identical distributions in both; labels show the share at 0.0010.001 . (b) Reassignment-minus-notation action distance across 1818 within-family traces. The marker gives the median and bootstrap 95%95\% interval; shading marks values below the 0.050.05 decision threshold.
Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

DimensionRange
optimiser{AdamW, Adam}
schedule{none, cosine, step}
base learning rate ηbase\eta_{\mathrm{base}}[10−5,10−2][10^{-5},10^{-2}]
block multipliers mlift,mspec,mbyp,mprojm_{\mathrm{lift}},m_{\mathrm{spec}},m_{\mathrm{byp}},m_{\mathrm{proj}} (4)[0.5,2.0][0.5,2.0]
weight decay[0,10−2][0,10^{-2}]
rollout length kk{1,2,4,8}\{1,2,4,8\}
Appendix
Table 3: Action dimensions and bounds shared by all policies. The four block multipliers are separate coordinates. Together, the rows define eight numeric and four categorical coordinates, matching D=12D=12 in Equation 1 .
ContextContrastTracesMedian gappp
Within-familyreassignment versus notation18+0.085+0.0850.0156250.015625
Within-familynotation versus resampling18+0.000+0.0000.630.63
Cross-familyreassignment versus resampling18+0.083+0.0830.0156250.015625
Appendix
Table 4: Action-distance summaries for the validation-history intervention. Exact one-sided pp -values use six cell medians. The cross-family result tests whether the effect extends to cross-family transfers.
CoordinateResamplingNotationReassignment
Rollout length kk14.4%14.4\%11.1%11.1\%49.6%49.6\%
Data loss15.9%15.9\%16.7%16.7\%40.4%40.4\%
Schedule14.1%14.1\%13.3%13.3\%25.9%25.9\%
Optimiser6.3%6.3\%7.0%7.0\%7.8%7.8\%
Appendix
Table 5: Within-family categorical flip rates relative to the logged configuration. No per-coordinate tests are applied.
Figure 3: Run-level endpoint performance and initial proposal quality. (a) Paired baseline-to-LLM held-out test nRMSE ratios for each cell and seed; values above one favour the LLM, and amber marks the single TPE win. (b) Placement of each first proposal within the pooled distribution of the 6060 random-search validation configurations from the same cell. Blue denotes within-family and magenta cross-family transfer. Panel (b) does not measure held-out test benefit.
Figure 4: Cumulative-best validation nRMSE as a function of trial number for all cells, with three seeds per method. Thin lines are individual runs; thick lines are medians across the three seeds. Panels (a) – (f) are the within-family cells. Of the cross-family panels, (g) – (i) adapt an advection-pretrained FNO to Burgers’ and (j) – (l) reverse the transfer.

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