PaperlayerView exact revision on arXiv

Source-linked AI summary

Recursive Improvement of a Differentiable Scientific Software Ecosystem

Pengcheng Hou, Xiaojun Tan, Sihan Hu, Ruisi Wang, Shuo Chen, Lei Wang, Youjin Deng, Kun Chen

arXiv:2610.04561v2cs.AIphysics.comp-ph

TL;DR

Scientific software offers established computations for gradient-based research, but extending differentiation across heterogeneous tools requires suitable interfaces and numerical evaluation. This paper builds an agent-driven ecosystem of reusable AD extensions and research programs. Across 20 measured gradient requests, AD is less costly than finite differences, while research feedback supports revisions to capabilities and scientific workflows.

  • Problem

    Extending differentiation across heterogeneous scientific software requires suitable boundaries and interfaces that preserve domain-specific meaning and numerical results.

    Source: §1 · pp. 1–2

  • Method

    The authors build an agent-driven ecosystem of reusable AD extensions, common differentiation conventions, research programs, and numerical references.

    Source: §1 · p. 2, §2.1 · p. 3

  • Results

    Across all 20 measured gradient requests, AD costs less than central FD; the 144-control QuantumOptics gradient takes 16.5 ms with AD and 4,570 ms with FD.

    Source: §3.2 · p. 4, §7 · p. 9

  • Takeaways & Limitations

    Research programs, checks, and saved outputs support further software validation and research, connecting task outcomes to continued development of shared resources.

    Source: §7 · p. 9

    Fourteen Figure 2 rows compare FD on fewer than all active coordinates, and their plotted times are not extrapolated to full-gradient costs.

    Source: §3.2 · p. 4

Abstract

from arXiv · show

Differentiable programming connects scientific computation with gradient-based inference, learning and design. Extending these capabilities across a heterogeneous software ecosystem requires specialized effort to implement derivatives, integrate interfaces and evaluate quality. AI coding agents can accelerate this transformation, but translating their capabilities into useful scientific software requires identifying research needs and evaluating how well implementations meet them. We present an environment for agent-driven evolution of differentiable scientific software that connects demand identification, development and quality evaluation. A unified differentiation interface exposes reusable derivative rules alongside existing numerical routines, allowing research tasks to share these capabilities. Research requirements guide development, with implementations assessed through independent derivative checks, workflow tests and performance evaluation. Validated software, research programs and tests become shared resources for subsequent studies. We construct and validate automatic differentiation extensions across 20 packages spanning physical, chemical and biological modeling, with research workflows demonstrating reuse across tasks. Benchmarks demonstrate computational savings over finite differences in gradient evaluation and complete parameter estimation. Research-driven revisions make previously unsupported workflows differentiable, correct derivatives of scientific observables and eliminate redundant computation. Quantum-control and thermal-design studies revise objectives in response to physical evaluation, improving designs while reusing existing derivatives. This work provides a practical approach to expanding differentiable programming across established scientific software and organizing AI agents around the recursive improvement of a shared computational ecosystem.

1 INTRODUCTION

Differentiable programming can connect established scientific computations to gradient-based learning, inference, and design, but extending it across heterogeneous software requires suitable differentiation boundaries, interfaces, and numerical checks. The paper proposes recursive improvement, in which research uses shared software and returns programs, checks, and requirements that guide further development.

1 INTRODUCTION

  • Extending differentiation across scientific software requires interfaces and derivative boundaries that preserve domain-specific meaning across data structures, solvers, and execution models.A derivative may need to follow a parameter’s effects through assembled equations, a solved field, and a measured flux.Source: §1 · pp. 1–2
  • Research programs expose requirements that code generation alone cannot establish, including missing observable dependencies and physical conditions absent from an objective.The authors identify research programs as concrete sources of requirements and tests for further development.Source: §1 · p. 2
  • Research uses shared software and returns implementations, programs, and checks that can reveal further needs and guide subsequent development.The authors call this connection recursive improvement.Source: §1 · p. 2
  • The ecosystem combines validated AD extensions across 20 scientific packages with common differentiation conventions, executable research programs, and numerical references.The collection spans molecular and cellular modeling, chemical kinetics, materials and quantum physics, and Earth and space sciences.Source: §1 · p. 2
  • The work demonstrates research-driven capability expansion, correction of scientific observables, reduced redundant computation, and application revisions informed by physical evaluation.Applications include more error-tolerant quantum control and thermal designs for additional heating conditions.Source: §1 · p. 3

2 A DIFFERENTIABLE SCIENTIFIC SOFTWARE ECOSYSTEM

The ecosystem connects reusable derivative capabilities to research programs through development, evaluation, and revision. Its interfaces expose numerical responses while research code defines task-specific inputs and objectives, and the resulting programs and checks feed subsequent development.

2 A DIFFERENTIABLE SCIENTIFIC SOFTWARE ECOSYSTEM

  • The ecosystem links reusable numerical capabilities to research programs through development, evaluation, and revision.Research programs, checks, and saved outputs remain resources for subsequent tasks.Source: §2 · p. 3

2.1 REUSABLE COMPUTATIONS AND DIFFERENTIATION INTERFACES

  • An AD extension exposes values and derivatives for specified continuous inputs within a documented domain, while research code supplies parameterization, observations, and loss.For y = f(θ; c), θ denotes active inputs and c fixed context; JVPs and VJPs describe numerical response.Source: §2.1 · p. 3
  • The shared differentiation abstraction has runtime-specific bindings rather than a single execution engine.The evaluated FiPy programs use custom Autograd rules, while QuantumOptics uses ChainRulesCore with Zygote.Source: §2.1 · p. 3

2.2 DEVELOPMENT GUIDED BY RESEARCH USE

  • Development agents inspect original software, propose computations and differentiation strategies, and implement rules with examples and numerical tests after review.Separate reviewer agents assess the proposed design before implementation.Source: §2.2 · p. 3
  • Research agents turn scientific questions and software examples into executable models and evaluation programs; checks cover value agreement, derivatives, scientific outcomes, and computational costs.Papers, public research code, and the Large Knowledge Model inform task discovery.Source: §2.2 · p. 3
  • Research feedback can carry a computation, fixed inputs, and numerical checks into a development request, then reuse the same inputs and outputs to test revisions.The Kwant example supplies an eight-gate device and gate-to-transmission checks; later physical evaluation can instead motivate revised objectives using existing derivatives.Source: §2.2 · p. 3

3 CAPABILITIES AND SCIENTIFIC UTILITY

The evaluated ecosystem covers reusable derivative capabilities and numerical checks across scientific tasks. AD costs less than finite differences in all 20 measured gradient requests, while task workflows compose shared derivatives for different scientific objectives.

3.1 REUSABLE CAPABILITIES AND NUMERICAL VALIDATION

  • Shared solves, propagations, and rule compositions support task-specific definitions, with research code selecting the observable or objective and packages supplying reusable derivatives.The passage gives quantum control, porous transport, and hydrologic sensitivities as examples.Source: §3.1 · p. 4
  • Numerical checks compare derivative computations with original-software or independent values, finite differences, analytic derivatives, and task-level references.Examples include a QuantumOptics 144-control gradient discrepancy of 1.16 × 10^-8 against central FD and a FiPy series reaching 1,024 conductivity parameters with maximum relative gradient discrepancy 8.46 × 10^-8.Source: §3.1 · p. 4

3.2 DERIVATIVE COSTS AND RESEARCH-TASK EFFICIENCY

  • AD is less costly than central FD in all 20 measured gradient requests; the 144-control QuantumOptics gradient takes 16.5 ms with AD and 4,570 ms with FD.AD can share work across gradient components, whereas central FD repeats forward evaluations at perturbed parameters.Source: §3.2 · p. 4, Figure 2 · p. 5
  • Fourteen rows compare FD on fewer than all active coordinates, and their timings are not extrapolated to full-gradient FD costs.For MCell, a model-specific expected-yield rule avoids the native Monte Carlo sampling used by FD at the prescribed accuracy.Source: §3.2 · p. 4
  • Matched synthetic FiPy and Kwant inversions use the same noiseless, same-model observations and bound-constrained optimizer; both methods recover parameters with similar accuracy.The reported fits also make similar predictions under an additional FiPy loading or at additional Kwant energies.Source: §3.2 · p. 4

4 RESEARCH-DRIVEN EVOLUTION OF SHARED CAPABILITIES

Research tasks reveal missing derivative paths, scientific-semantic errors, and redundant computation in shared capabilities. The resulting changes support evaluated designs and reuse of research inputs and checks in later software revisions.

4.1 ITERATIVE SOFTWARE IMPROVEMENT THROUGH RESEARCH TASKS

  • A research task specified the missing Kwant gate-to-transmission derivative path, and a new implicit scattering rule passed independent task-level retesting.Using that rule, the AD design meets reserved-energy mean stop/pass transmission thresholds of at most 0.5 and at least 1.5, with means of 0.266 and 1.737.Source: §4.1 · p. 5
  • A Kwant revision shares the scattering-system linearization and adds an adjoint, reducing derivative factorizations from 48 to six for one six-energy value-and-gradient request.At the saved design, median time falls from 1.34 to 0.48 s, compared with 0.88 s for central FD; these timings compare fixed inputs, not complete design runs.Source: §4.1 · pp. 5–6, Figure 3 · p. 6

4.2 RESEARCH-DRIVEN IMPROVEMENTS ACROSS DOMAINS

  • In StochKit2, the old rule returned [−0.5, 0.5] instead of the reference [0, 0] because both derivative modes recomputed automatic bin edges rather than using requested bins.The repair carries requested bins through both modes and agrees with an independent reconstruction; a later revision reduces duplicate native calls from four to two with bitwise-identical outputs.Source: §4.2 · p. 6
  • Research workflows also add OpenPNM transport derivatives, repair BioDynaMo mechanics updates, and consolidate QuSpin propagation derivatives into a shared package rule.The QuSpin rule later supplies both control objectives, and research definitions become numerical checks with saved inputs and outputs supporting later changes.Source: §4.2 · p. 6

5 EVOLVING RESEARCH APPLICATIONS

The research applications reuse existing differentiation capabilities while revising objectives in response to physical evaluation. The revised quantum-control and thermal-cloak tasks improve performance under targeted conditions, while also exposing trade-offs and conditions where performance remains limited.

5.1 ROBUST QUANTUM CONTROL AND PHASE SENSING

  • QuSpin propagation derivatives supply gradients for nominal and coupling-aware objectives for a four-spin GHZ-state preparation using 96 bounded control amplitudes.The revised objective averages preparation error at three coupling values; the paired objective comparison uses equal budgets of 660 single-scenario value-and-gradient evaluations.Source: §5.1 · p. 7
  • Worst error 1 − F falls from 10.2–15.8% to 0.020–0.090% after the quantum-control objective is revised for coupling variation.The improvement comes with larger root-mean-square control amplitudes and a small loss of nominal fidelity; all six runs stop at their prescribed budgets.Source: §5.1 · p. 8
  • Saved states from both control objectives retain sensing utility close to the ideal GHZ reference, with 2.48–2.86 milliradian RMS errors across two phases.The subsequent sensing test uses a fixed, known +10% coupling error and the same ideal readout and estimator, without another pulse optimization.Source: §5.1 · p. 8

5.2 THERMAL-CLOAK DESIGN UNDER MULTIPLE HEATING CONDITIONS

  • Cubic RMS disturbance relative to the bare core falls from 2.96 to 0.00371 on a common 192^2 evaluation grid after adding that load and refining training.The revised ratio is 0.0446 at 576^2.Source: §5.2 · p. 8, Figure 5 · p. 8
  • At 576^2, the fixed revised cloak reduces disturbance by 95.5–99.2% on three design loads and 96.9% under a new localized hotspot, but a quartic profile yields 2.78 times bare-core disturbance.The quartic grid-refinement response fails the prespecified stability check, and analytic uniform and optimized two-layer coatings each achieve lower disturbance on individual loads; the revised design is a multiload compromise.Source: §5.2 · p. 8, Figure 5 · p. 8

6 RELATED WORK

Related work spans differentiable computing, agents for software and scientific research, and evaluator-driven program improvement. This paper builds on these areas by following reusable numerical software across research tasks and examining how task requirements revise shared implementations.

Differentiable scientific computing

  • Established AD systems and solver-boundary rules support differentiable simulation, while ChainRules separates derivative definitions from engines and benchmarks compare implementations across languages and workloads.The paper examines whether derivatives preserve scientific meaning across research uses and how task requirements change shared implementations and checks.Source: §6 · p. 9

Agents for software and scientific research

  • Prior agent work studies repository repair, scientific coding and research extensions, while this work follows reusable numerical software as use changes its implementation and validated results support later studies.ToolMaker builds agent-callable tools from papers with code; Lahlou et al. translate a legacy Fortran model into JAX with a numerical reference; ScienceIDE uses scientific repositories for agent learning and evaluation.Source: §6 · p. 9

Evaluator-driven and recursive improvement

  • Unlike evaluator-guided program search or systems that modify the agent, this work improves scientific software and its research programs through evaluation and reuse of prior results.Evaluation can change a numerical capability or scientific objective, and a preceding research result can become an input to later software validation.Source: §6 · p. 9

7 CONCLUSION

The paper builds an agent-driven ecosystem for differentiable scientific software and studies how research feedback improves shared capabilities and their uses. Numerical checks and physical evaluation inform revisions, while executable programs, checks and saved outputs support subsequent work.

7 CONCLUSION

  • Across 20 scientific packages, numerical checks validate diverse derivative computations, and measured gradient requests cost less than their finite-difference comparisons.Research studies show support for inference and design, while feedback adds missing dependencies, corrects scientific semantics and reduces repeated computation in shared implementations.Source: §7 · p. 9
  • Scientific checks test whether a computation matches its intended observable, while physical evaluation tests whether it serves the research objective under relevant conditions.These distinct questions motivate revisions to shared implementations or to the objectives built on them.Source: §7 · p. 9
  • The broader opportunity is to use AI agents to reshape scientific software ecosystems, with differentiation connecting established computations to learning, inference and design.The paper identifies uncertainty quantification and coupled simulation as other capabilities that could be connected, and accumulating implementations and validation knowledge as a route to resources supporting wider research.Source: §7 · p. 9

REPRODUCIBILITY STATEMENT

The appendices describe development procedures, differentiation interfaces, evaluated computations, numerical checks and timing procedures. The supplementary archive includes records, comparisons, task programs and portable checks, with reanalysis distinguished from fresh reruns.

REPRODUCIBILITY STATEMENT

  • Appendices A–F document agent development and interfaces, evaluated computations and timing, software revisions and retests, and research-task models and settings.Source: REPRODUCIBILITY STATEMENT · p. 10
  • The supplementary archive provides numerical records, componentwise AD/FD comparisons, task programs and portable verification checks.Source: REPRODUCIBILITY STATEMENT · p. 10
  • The archive records software versions and runtime requirements and distinguishes reanalysis of saved records from fresh scientific reruns with package-specific environments.The paper and appendices define the reported inputs, comparisons and evaluation conditions.Source: REPRODUCIBILITY STATEMENT · p. 10

AI USE STATEMENT

Generative AI tools supported both software development and research activities, from investigation and implementation to manuscript preparation.

AI USE STATEMENT

  • Generative AI tools supported software investigation, AD-interface design, implementation, testing, and revision, as well as literature discovery, study design, experiments, analysis, and manuscript preparation.Synthetic observations came from numerical simulation programs developed with agent assistance; validation included package tests and comparisons with independent numerical references.Source: AI USE STATEMENT · p. 10

A.1 DEVELOPMENT ROLES AND STAGE OUTPUTS

The appendix describes staged agent development, research-workflow discovery, and the differentiation interfaces and evaluated capabilities used in the experiments. The procedure combines software-specific derivative rules with research programs and numerical checks.

A.1 DEVELOPMENT ROLES AND STAGE OUTPUTS

  • The four-stage development procedure separates investigation and specification, independent design review, revision, and implementation, with documented outputs passed between stages.A retained QuSpin review changed the derivative target for hamiltonian.evolve to distinguish initial-state derivatives from Hamiltonian derivatives; implementation reported 128 of 130 targets, with two original-call failures unresolved. Optional delivery review and repeated design reviews are not documented for every package.Source: §A.1 · p. 16, Table 2 · p. 16

A.2 WORKFLOW DISCOVERY AND EVALUATION

  • Research agents use literature, public code, original-software examples, and source-grounded LKM reasoning graphs to construct executable tasks and evaluate numerical results and costs.The original code and specified inputs, not the reasoning graph alone, define the computation; selected case counts do not estimate development success rates.Source: §A.2 · p. 16

A.3 DIFFERENTIATION INTERFACES USED IN THE EXPERIMENTS

  • The experiments use runtime-specific bindings for explicit derivative rules, including Python ChainRules, Autograd primitives, and Julia ChainRulesCore composed through Zygote.For linear solves, a reusable solve rule supplies the derivative contribution while task code adds the direct parameter derivative of its loss, so different observables can share the rule.Source: §A.3 · p. 17

A.4 COMPOSING A STEADY-STATE DERIVATIVE

  • The QuantumOptics benchmark composes a package steady-state derivative rule with task-level reverse mode to evaluate full-gradient requests for a 12-dimensional system at control counts from 1 to 144.The benchmark tests the composed gradient, not a completed inverse-design optimization.Source: §A.4 · p. 17

B EVALUATED CAPABILITIES, NUMERICAL VALIDATION AND COST

  • The gradient-cost collection covers delivered derivative paths across 20 packages, with one specified workload per package for the Figure 2 comparison.Research applications also use some paths in larger computations.Source: B EVALUATED CAPABILITIES, NUMERICAL… · p. 17
  • Table 3 catalogs evaluated parameter-to-output paths and numerical checks across 20 packages; MCell’s rate-amplitude rule was checked against native Monte Carlo finite differences under a 10% gradient-error screen.These evaluated paths are not complete API inventories.Source: Table 3 · p. 18, B EVALUATED CAPABILITIES, NUMERICAL… · p. 18

B.1 BROADER WORKFLOW CAPABILITIES AND RULE REUSE

The section examines reuse of shared differentiation rules across scientific workflows, alongside gradient costs, numerical checks, and the limits of performance and precision claims. It also shows how independent research requirements prompted tested revisions that corrected derivative behavior and reduced redundant computation in Kwant and StochKit2.

B.1 BROADER WORKFLOW CAPABILITIES AND RULE REUSE

  • Workflow reports span transport, quantum, molecular and cellular, chemical and stochastic, and spectral and gravitational computations, with derivatives bounded to specified paths and conditions.Examples include fixed structures or visibility branches, fixed states or grids, and supported simulator or response paths rather than arbitrary algorithms or discrete events.Source: §B.1 · pp. 19–20

B.2 SCALAR OBJECTIVES AND DIFFERENTIATION COORDINATES

  • Each gradient workload fixes a scalar objective and active inputs; coordinate choices include 528 independent real QuSpin coefficients and 1,024 packed-complex SHTns degrees of freedom.Other perturbations use ambient force-constant arrays or positive TeNPy vector entries without enforcing symmetry, acoustic constraints, or simplex renormalization.Source: §B.2 · p. 20

B.3 NUMERICAL CHECKS

  • Maximum scaled differences were 3.81 × 10−10 for FiPy, 5.12 × 10−8 for EvoEF2, and 5.09 × 10−7 for PHREEQC; QuantumOptics’ relative gradient L2 difference was 1.16 × 10−8.These checks assess defined callables and are not interchangeable measures of whole-package accuracy.Source: §B.3 · p. 22

B.4 TWENTY-PACKAGE GRADIENT REQUESTS AND TIMINGS

  • Fourteen workload rows compare selected-coordinate FD with an AD call returning the complete gradient, while other rows request all task-level coordinates.Timing scopes differ: BioDynaMo and PhysiCell time reimplemented diffusion kernels, and MCell compares an analytical expectation rule with native Monte Carlo FD.Source: §B.4 · pp. 22–24

B.5 COMPLETE-GRADIENT SCALING AND COMPUTATIONAL WORK

  • At 1,024 parameters, FiPy takes 6.66 ms with AD versus 7.90 s with FD, while Kwant takes 34.2 ms versus 9.56 s.FiPy keeps its 32 × 32 mesh fixed; Kwant grows to N = P + 4 sites at energy 0.23. These are complete value-and-gradient requests.Source: Figure 7 · p. 24, §B.5 · p. 24

B.6 MCELL: EXPECTED-YIELD SENSITIVITY AT A FIXED ACCURACY TARGET

  • 0.03896 s for AD versus 71.6914 s for FD yields an approximately 1840× ratio for MCell’s one-parameter expected-yield sensitivity.The comparison uses a model-specific analytical expectation rule and native Monte Carlo FD at the stated precision; all seven primary and validation requests pass their checks.Source: §B.6 · p. 25

C RESEARCH-DRIVEN SOFTWARE REVISIONS

  • Research-driven cases connect findings to changed implementations or compositions, with direct before-and-after executions for Kwant and StochKit2 and workflow reports specifying tested versions, inputs, observables, and checks.Case counts describe tested tasks, not fractions of a package’s complete API.Source: C RESEARCH-DRIVEN SOFTWARE REVISIONS · p. 25

C.1 KWANT TRANSPORT EXTENSION AND FILTER DESIGN

  • Kwant added an implicit derivative for its native scattering solver, and both AD and FD designs reduced training loss from 0.49921 to 0.02082 and 0.02107, respectively.Both designs met reserved-energy criteria; the tested-point maximum flux-conservation residual was 4.00 × 10−15, not a continuous-band guarantee.Source: §C.1 · pp. 26–27, Figure 8 · p. 27
  • At 21 sampled points per specified band, both final Kwant designs met the stop- and pass-band transmission thresholds, but the grid does not guarantee behavior between samples.All 303 native scattering calls completed with two channels per lead and maximum flux residual 1.35 × 10−14; the study evaluated saved designs without reoptimization or selection.Source: §C.1 · pp. 27–28
  • System assemblies fell from 54 to 12 and derivative factorizations from 48 to six; the revised Kwant call cost about 2.8× less than the earlier implementation and 1.83× less than same-environment finite differences.Full gradients agreed between versions to at most 7.15 × 10−16, and independent finite-difference checks passed; the timing comparison is bounded to an eight-parameter task, not a whole-optimizer speedup.Source: §C.1 · p. 28

C.2 STOCHKIT2: FIXED-BIN DERIVATIVES AND SIMULATION REUSE

  • The StochKit2 histogram derivative was wrong when automatic bin bounds replaced the requested grid; revision 9d098b26 forwards and validates the fixed grid, correcting the tested probability vector and derivative.For terminal populations [0, 2] and supplied scores [−1, 1], the correct fixed-bin probability vector is [1, 0] and its derivative is [0, 0], whereas the old rule returned [−0.5, 0.5] despite passing JVP/VJP duality.Source: §C.2 · p. 28
  • The four-seed mean’s relative gradient L2 error fell from 0.732 to 0.199 as trajectory count increased from 8,192 to 131,072, while larger-run single-seed errors remained 0.219–0.532.One of three initial 512-trajectory ensembles failed the predeclared analytic-gradient screen because an empty bin had zero empirical derivative and standard error despite a nonzero exact derivative; four replicates do not establish interval coverage or completed parameter fitting.Source: §C.2 · p. 28
  • StochKit2’s fixed-grid repair eliminated duplicate seeded simulation: actual native calls fell from four to two, with bitwise-identical value and derivative arrays matching an independent reconstruction.The comparison used a native pure-death model with 20,000 realizations and a fixed seed; no distribution-fit or statistical-accuracy claim follows, and reported medians include instrumentation rather than isolating speedup.Source: §C.2 · p. 29

C.3 FIPY EXECUTION CHANGES AND ADDITIONAL IMPLEMENTATION EVIDENCE

The FiPy execution revision combines preparation reuse with reverse-mode task code, while workflow-driven changes expand or repair differentiable capabilities across packages. The four-spin study uses shared QuSpin derivatives to prepare and evaluate GHZ controls.

C.3 FIPY EXECUTION CHANGES AND ADDITIONAL IMPLEMENTATION EVIDENCE

  • For FiPy’s molecular-messenger workflow, AD median runtime fell from 3.816 to 1.408 seconds after combined package-preparation reuse and reverse-mode task-code revisions.The comparison does not isolate either change or numerical-factorization reuse.Source: §C.3 · p. 29
  • Figure 9 compares ten FiPy workloads, retaining four later-AD medians slower than FD and Shape*, whose derivative check remains unresolved.Shape* timing supports an execution-cost comparison, not independently verified derivative accuracy.Source: Figure 9 · p. 29, §C.3 · pp. 29–30

C.4 WORKFLOW-DRIVEN CAPABILITY REPAIR AND EXPANSION

  • QuSpin’s shared propagation rule increased evaluated passing programs from 23 to 25 of 47, including new dynamics-learning Jacobian and gate-synthesis directional-derivative checks.The later four-spin control study uses the shared Hamiltonian and input-state pullbacks.Source: §C.4 · p. 30
  • OpenPNM’s checked transport cases increased from 42 to 66, adding reaction–diffusion and advection–diffusion derivatives while 14 hard-threshold capillary-invasion cases remained unsupported.The added derivatives cover reaction-source arrays and directed-flow components.Source: §C.4 · p. 30
  • QuantumOptics added nine checked compositions of existing rules, increasing passes from 35 to 44 of 49 tasks; five tasks remained unsupported.The maximum reported independent-FD discrepancy was 1.75 × 10^-7.Source: §C.4 · pp. 30–31

D QUANTUM CONTROL AND PHASE ESTIMATION

  • The four-spin optimizations reached fidelity 0.9999947348–0.9999999653, but all stopped at the iteration limit and derivative checks sampled directions rather than every coordinate.The study did not measure an AD/FD optimization-time ratio.Source: D QUANTUM CONTROL AND PHASE ESTIMATION · p. 31
  • The representative control’s half-chain purity changed from 1 to 0.4999998199, with S2 approaching one bit; GHZ overlap, not entropy alone, identifies the target state.Figure 10 records its 96 control amplitudes, nominal fidelity and entropy evolution, and fixed-pulse responses across the coupling-error grid.Source: D QUANTUM CONTROL AND PHASE ESTIMATION · p. 32, Figure 10 · p. 32

D.1 REVISING THE CONTROL OBJECTIVE AFTER CALIBRATION-ERROR EVALUATION

The control study revises its objective to average three observed coupling-error scenarios, then evaluates the resulting designs on a separate confirmation grid. A fixed-readout study examines phase information and selected finite-shot estimation outcomes, while the following material-error study applies a further objective revision.

D.1 REVISING THE CONTROL OBJECTIVE AFTER CALIBRATION-ERROR EVALUATION

  • The revised objective averages three previously observed coupling-error scenarios rather than minimizing the worst case, and the confirmation points were fixed before re-optimization.The six runs received equal budgets of 660 single-scenario value-and-gradient evaluations, but exhausted them after different iteration counts.Source: §D.1 · pp. 32–33
  • Minimum confirmation fidelities were 0.898473, 0.842458 and 0.882875 for nominal controls versus 0.999657, 0.999797 and 0.999100 for three-scenario designs, with higher control RMS.The designs were paired by initialization and assessed at ten fixed confirmation points; these finite-grid results do not guarantee performance across continuous errors.Source: §D.1 · p. 33, Figure 11 · p. 33

D.2 FIXED-READOUT FINITE-SHOT PHASE ESTIMATION

  • On the evaluated information grid, robust controls remained above 0.9983 in FC/FQ, while the nominal minimum was 0.3114.The study distinguishes information in the state from information accessible through the chosen fixed readout.Source: §D.2 · p. 34
  • At 10,000 measurements, the two prepared states’ phase errors were roughly 2.5–2.9 milliradians, near the ideal GHZ reference and below the unentangled reference’s 4.8–5.1 milliradians.The nominal-to-robust RMSE reduction was 13.4% at one tested phase and 2.5% at the other; the latter was smaller than the combined Monte Carlo standard error, so this selected comparison does not establish a general gain.Source: §D.2 · p. 34

E FIPY CLOAK DESIGN AND MATERIAL-ERROR ANALYSIS

E.1 PASSIVE MULTILOAD DESIGN AND TASK REVISION

Across thermal design and synthetic inversions, research evaluation exposed limits of particular designs while showing that AD could reduce computational cost with similar recovery and prediction quality. The studies also illustrate how objectives and checks shape the conclusions that can be drawn from each workflow.

E.1 PASSIVE MULTILOAD DESIGN AND TASK REVISION

  • Adding the cubic load to training changed its 192² cubic ratio from 2.956 times bare-inclusion disturbance to 0.003713, but this was a repaired training condition, not unseen-load generalization.The training grid and objective changed together, so their separate effects cannot be attributed.Source: §E.1 · p. 35
  • At 576² cells, exterior RMS disturbance reductions were 97.05%, 99.20%, and 95.54% for horizontal, quadratic, and cubic loads, but the design was a constrained multiload compromise.The quartic challenge had 2.777 times the bare-inclusion disturbance; the uniform coating did better for horizontal loading and the two-layer baseline for cubic loading.Source: §E.1 · p. 35

E.2 MATERIAL-ERROR FOLLOW-UP

  • The material-error study’s training-grid worst-scenario loss fell by about 5%, but the design ordering reversed on the finer grid; the frozen, budget-limited comparison cannot isolate the cause.The study tests a three-load mean across five material scenarios, not the worst individual load or the whole error box.Source: §E.2 · pp. 35–36

F.1 THERMAL CONDUCTIVITY INVERSION

  • AD and FD recovered conductivity with relative L2 errors of 3.07 × 10−6 and 3.46 × 10−6; the complete-process FD/AD time ratio was 28.6.This was a same-grid synthetic inversion using full-field observations, not a noisy or sparse-sensor test.Source: §F.1 · p. 37, Figure 12 · p. 37

F.2 LOCAL-DENSITY-OF-STATES INVERSION

  • Kwant’s AD and FD fits reached final losses of approximately 6.72×10−10, with AD parameter-recovery relative error 1.33×10−4 and withheld-output RMSE 3.87×10−7.The withheld prediction covered 72 site/energy values; parameters use the benchmark’s numerical energy convention, not specified material units.Source: §F.2 · pp. 37–38

F.3 PREDICTION, RECOVERY AND OPTIMIZATION COST

  • FD/AD optimization-time ratios were 39.5 in FiPy and 29.5 in Kwant, with similar recovery and prediction quality on the specified synthetic problems.Figure 14 records FiPy objective trajectories and Kwant total optimizer times; Kwant iteration-level times were not recorded.Source: §F.3 · pp. 38–39, Figure 14 · p. 39

Related papers

Loading 2610.04561v2…