pith. sign in

arxiv: 2606.25502 · v1 · pith:PGNEY6WWnew · submitted 2026-06-24 · ❄️ cond-mat.mtrl-sci · physics.comp-ph

A Differentiable DFT-Based Framework for Inverse Materials Design

Kohei Ishii , Hisazumi Akai , Tetsuya Fukushima , Hikari Shinya , Koji Inui This is my paper

Pith reviewed 2026-06-25 20:57 UTC · model grok-4.3

classification ❄️ cond-mat.mtrl-sci physics.comp-ph
keywords inverse materials designKKR-CPAautomatic differentiationgradient-based optimizationcompositional optimizationmagnetic alloyshalf-metals
0
0 comments X

The pith

Reverse-mode automatic differentiation makes KKR-CPA gradients with respect to composition independent of the number of candidate elements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper integrates reverse-mode automatic differentiation directly into the KKR-CPA electronic-structure method so that atomic compositions become continuous variables that can be optimized by gradient descent. Because the gradient cost does not grow with the number of possible elements, the approach can search compositional spaces containing dozens of species. Any computable physical quantity can serve as the objective function. The framework is demonstrated on magnetic alloys and half-metals, returning explicit candidate compositions such as (Lu0.553Yb0.447)(Co0.759Fe0.241)2Fe3 and FeZr(Sb0.94Te0.06). The method therefore supplies a direct, first-principles route from a target property to the material that realizes it.

Core claim

By embedding reverse-mode automatic differentiation inside KKR-CPA, the authors obtain gradients of any objective function with respect to site compositions at a computational cost independent of the number of candidate elements; this enables gradient-based optimization over continuous composition variables that can later be discretized to realizable atomic arrangements.

What carries the argument

Reverse-mode automatic differentiation applied to the KKR-CPA method, with atomic compositions treated as continuous optimization variables.

If this is right

  • Any computable quantity can be used as the objective function for the optimization.
  • Compositional spaces spanning dozens of elements become tractable for gradient-based search.
  • The same differentiable pipeline applies to both magnetic alloys and half-metals without method changes.
  • Optimized continuous compositions can be discretized to produce concrete candidate formulas.
  • The framework supplies a physically grounded inverse-design route rather than exhaustive screening.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same reverse-mode approach could be ported to other electronic-structure codes once they expose differentiable quantities.
  • Continuous-composition optima might serve as starting points for subsequent discrete structure searches or cluster-expansion models.
  • Because the gradient cost is independent of element count, the method scales naturally to high-entropy alloys.
  • The framework could be combined with experimental feedback loops by using measured properties as objectives.

Load-bearing premise

KKR-CPA remains accurate enough when compositions are relaxed to continuous variables and the resulting values are later discretized back into realizable atomic arrangements.

What would settle it

An explicit calculation in which an optimized continuous composition, after rounding to integer occupancies, fails to reproduce the target property within the documented accuracy of KKR-CPA.

read the original abstract

Discovering solid-state materials with target properties remains a central challenge in computational materials science. Existing approaches -- high-throughput screening, surrogate optimization, and generative models -- require extensive evaluations or training data and extrapolate poorly to unseen compositions. Here we develop a first-principles inverse-design framework, integrating reverse-mode automatic differentiation (AD) into KKR-CPA -- the Korringa--Kohn--Rostoker method with the coherent potential approximation -- where atomic compositions are continuous variables to be optimized. Reverse-mode AD yields gradients of objective functions with respect to composition at a cost independent of the number of candidate elements, enabling gradient-based optimization to identify materials from compositional spaces spanning dozens of elements. In this framework, any computable quantity can serve as the objective. We demonstrate this generality through two contrasting applications, magnetic alloys and half-metals, yielding candidates such as (Lu$_{0.553}$Yb$_{0.447}$)(Co$_{0.759}$Fe$_{0.241}$)$_2$Fe$_3$ and FeZr(Sb$_{0.94}$Te$_{0.06}$). Our framework offers a physically grounded route from a target property to the material that realizes it.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The manuscript develops a first-principles inverse materials design framework by integrating reverse-mode automatic differentiation into the KKR-CPA electronic structure method, treating atomic compositions as continuous variables that can be optimized via gradients whose computational cost is independent of the number of candidate elements. The approach is demonstrated on two applications: magnetic alloys and half-metals, producing candidate materials such as (Lu_{0.553}Yb_{0.447})(Co_{0.759}Fe_{0.241})_2Fe_3 and FeZr(Sb_{0.94}Te_{0.06}).

Significance. If the results hold, the framework offers a scalable, physically grounded method for inverse design that avoids the need for large training datasets or exhaustive screening, with the key advantage of gradient costs independent of composition space dimensionality. The integration of AD with an established first-principles method like KKR-CPA and the generality to any computable objective function are notable strengths.

major comments (1)
  1. [Abstract] The validity of the framework depends on KKR-CPA providing accurate gradients and optima when compositions are treated as continuous variables and subsequently discretized to integer occupations. The reported candidates use fractional compositions (e.g., Lu_{0.553}Yb_{0.447}), but the manuscript provides no evidence that these discretized structures recover the optimized properties when evaluated with supercell calculations or compared to experiment.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their constructive comments. We provide a point-by-point response to the single major comment below.

read point-by-point responses

  1. Referee: [Abstract] The validity of the framework depends on KKR-CPA providing accurate gradients and optima when compositions are treated as continuous variables and subsequently discretized to integer occupations. The reported candidates use fractional compositions (e.g., Lu_{0.553}Yb_{0.447}), but the manuscript provides no evidence that these discretized structures recover the optimized properties when evaluated with supercell calculations or compared to experiment.

    Authors: KKR-CPA is specifically formulated to treat atomic compositions as continuous variables that represent disordered (random) alloys through the coherent potential approximation. The optimization in our framework occurs entirely within this continuous representation, and the reported compositions (such as Lu_{0.553}Yb_{0.447}) are the direct outputs of the gradient-based procedure. All properties are evaluated consistently inside the same KKR-CPA model. Discretization to integer occupations would instead describe ordered supercell structures, which constitute a physically distinct system (ordered versus random alloy) and would require an entirely separate computational approach. Because the framework is defined and validated within the CPA for continuous compositions, such supercell comparisons lie outside its scope and are not needed to establish its validity. revision: no

Circularity Check

0 steps flagged

No circularity: AD integration is a standard technique applied to external KKR-CPA solver

full rationale

The paper presents a new integration of reverse-mode automatic differentiation into the existing KKR-CPA electronic-structure method to enable gradient-based optimization over continuous composition variables. No derivation step reduces by construction to a fitted parameter, self-definition, or self-citation chain; the cost-independence of reverse-mode AD is a general property of the algorithm and is not derived from the target result. The framework treats KKR-CPA as an external black-box oracle whose outputs are differentiated, with no evidence that any claimed prediction is statistically forced by the inputs or that a uniqueness theorem from the authors' prior work is invoked to close the argument. The central claim therefore remains self-contained against external first-principles benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that KKR-CPA supplies reliable property values under continuous composition relaxation; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption KKR-CPA provides sufficiently accurate electronic-structure predictions for alloy properties under the coherent potential approximation
    The entire optimization framework is built on top of this established method; its accuracy is presupposed for the inverse-design results to be meaningful.

pith-pipeline@v0.9.1-grok · 5758 in / 1338 out tokens · 28558 ms · 2026-06-25T20:57:07.915972+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

68 extracted references · 1 linked inside Pith

  1. [1]

    Mater.12, 191–201 (2013)

    Curtarolo, S.et al.The high-throughput highway to computational materials design.Nat. Mater.12, 191–201 (2013)

    2013

  2. [2]

    V., Xue, D

    Lookman, T., Balachandran, P. V., Xue, D. & Yuan, R. Active learning in mate- rials science with emphasis on adaptive sampling using uncertainties for targeted design.npj Comput. Mater.5, 21 (2019)

    2019

  3. [3]

    & Aspuru-Guzik, A

    S´ anchez-Lengeling, B. & Aspuru-Guzik, A. Inverse molecular design using machine learning: Generative models for matter engineering.Science361, 360–365 (2018)

    2018

  4. [4]

    Schmidt, J., Marques, M. R. G., Botti, S. & Marques, M. A. L. Recent advances and applications of machine learning in solid-state materials science.npj Comput. Mater.5, 83 (2019)

    2019

  5. [5]

    Himanen, L., Geurts, A., Foster, A. S. & Rinke, P. Data-driven materials science: Status, challenges, and perspectives.Adv. Sci.6, 1900808 (2019)

    2019

  6. [6]

    Inverse design in search of materials with target functionalities.Nat

    Zunger, A. Inverse design in search of materials with target functionalities.Nat. Rev. Chem.2, 0121 (2018)

    2018

  7. [7]

    arXiv:2509.07180

    Tahmassebpur, J.et al.Effective atom theory: Gradient-driven ab initio materials design (2025). arXiv:2509.07180

    arXiv 2025

  8. [8]

    G., Pearlmutter, B

    Baydin, A. G., Pearlmutter, B. A., Radul, A. A. & Siskind, J. M. Automatic differentiation in machine learning: a survey.J. Mach. Learn. Res.18, 1–43 (2018). URL https://jmlr.org/papers/v18/17-468.html

    2018

  9. [9]

    & Walther, A.Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation2nd edn (SIAM, Philadelphia, 2008)

    Griewank, A. & Walther, A.Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation2nd edn (SIAM, Philadelphia, 2008)

    2008

  10. [10]

    Photonics12, 659–670 (2018)

    Molesky, S.et al.Inverse design in nanophotonics.Nat. Photonics12, 659–670 (2018)

    2018

  11. [11]

    & Motome, Y

    Inui, K. & Motome, Y. Inverse Hamiltonian design by automatic differentiation. Commun. Phys.6, 37 (2023)

    2023

  12. [12]

    & Motome, Y

    Inui, K. & Motome, Y. Inverse Hamiltonian design of highly entangled quantum systems.Phys. Rev. Res.6, 033080 (2024)

    2024

  13. [13]

    & Saitoh, E

    Hirasaki, Y., Inui, K. & Saitoh, E. Inverse magnetoconductance design by automatic differentiation.Phys. Rev. B110, 214201 (2024)

    2024

  14. [14]

    Mater.9, 121 (2023)

    Liu, H.et al.End-to-end differentiability and tensor processing unit computing to accelerate materials’ inverse design.npj Comput. Mater.9, 121 (2023). 20

    2023

  15. [15]

    & Aspuru-Guzik, A

    Tamayo-Mendoza, T., Kreisbeck, C., Lindh, R. & Aspuru-Guzik, A. Auto- matic differentiation in quantum chemistry with applications to fully variational Hartree–Fock.ACS Cent. Sci.4, 559–566 (2018)

    2018

  16. [16]

    F., Lehtola, S

    Kasim, M. F., Lehtola, S. & Vinko, S. M. DQC: A Python program package for differentiable quantum chemistry.J. Chem. Phys.156, 084801 (2022)

    2022

  17. [17]

    Kasim, M. F. & Vinko, S. M. Learning the exchange-correlation functional from nature with fully differentiable density functional theory.Phys. Rev. Lett.127, 126403 (2021)

    2021

  18. [18]

    F., Ploumhans, B

    Schmitz, N. F., Ploumhans, B. & Herbst, M. F. Algorithmic differentiation for plane-wave DFT: materials design, error control and learning model parameters. npj Comput. Mater.12, 6 (2026)

    2026

  19. [19]

    Schoenholz, S. S. & Cubuk, E. D. JAX, M.D. a framework for differentiable physics.J. Stat. Mech. Theory Exp.2021, 124016 (2021)

    2021

  20. [20]

    Zunger, A., Wei, S.-H., Ferreira, L. G. & Bernard, J. E. Special quasirandom structures.Phys. Rev. Lett.65, 353–356 (1990)

    1990

  21. [21]

    Zur Elektronentheorie der Metalle

    Nordheim, L. Zur Elektronentheorie der Metalle. I.Ann. Phys.401, 607–640 (1931)

    1931

  22. [22]

    & Vanderbilt, D

    Bellaiche, L. & Vanderbilt, D. Virtual crystal approximation revisited: Applica- tion to dielectric and piezoelectric properties of perovskites.Phys. Rev. B61, 7877–7882 (2000)

    2000

  23. [23]

    On the calculation of the energy of a Bloch wave in a metal.Physica 13, 392–400 (1947)

    Korringa, J. On the calculation of the energy of a Bloch wave in a metal.Physica 13, 392–400 (1947)

    1947

  24. [24]

    & Rostoker, N

    Kohn, W. & Rostoker, N. Solution of the Schr¨ odinger equation in periodic lattices with an application to metallic Lithium.Phys. Rev.94, 1111–1120 (1954)

    1954

  25. [25]

    Coherent-potential model of substitutional disordered alloys.Phys

    Soven, P. Coherent-potential model of substitutional disordered alloys.Phys. Rev.156, 809–813 (1967)

    1967

  26. [26]

    & Ehrenreich, H

    Veliˇ ck´ y, B., Kirkpatrick, S. & Ehrenreich, H. Single-site approximations in the electronic theory of simple binary alloys.Phys. Rev.175, 747–766 (1968)

    1968

  27. [27]

    Gyorffy, B. L. Coherent-potential approximation for a nonoverlapping-muffin- tin-potential model of random substitutional alloys.Phys. Rev. B5, 2382–2384 (1972)

    1972

  28. [28]

    Fast Korringa-Kohn-Rostoker coherent potential approximation and its application to FCC Ni-Fe systems.J

    Akai, H. Fast Korringa-Kohn-Rostoker coherent potential approximation and its application to FCC Ni-Fe systems.J. Phys. Condens. Matter1, 8045–8064 (1989). 21

    1989

  29. [29]

    & Min´ ar, J

    Ebert, H., K¨ odderitzsch, D. & Min´ ar, J. Calculating condensed matter properties using the KKR-Green’s function method—recent developments and applications. Rep. Prog. Phys.74, 096501 (2011)

    2011

  30. [30]

    S., Stocks, G

    Faulkner, J. S., Stocks, G. M. & Wang, Y.Multiple Scattering Theory: Electronic Structure of Solids(IOP Publishing, Bristol, 2018)

    2018

  31. [31]

    M., Temmerman, W

    Stocks, G. M., Temmerman, W. M. & Gyorffy, B. L. Complete solution of the Korringa-Kohn-Rostoker Coherent-Potential-Approximation equations: Cu-Ni alloys.Phys. Rev. Lett.41, 339–343 (1978)

    1978

  32. [32]

    & Akai, H

    Takahashi, C., Ogura, M. & Akai, H. First-principles calculation of the Curie temperature Slater–Pauling curve.J. Phys. Condens. Matter19, 365233 (2007)

    2007

  33. [33]

    & Khmelevskyi, S

    Kudrnovsk´ y, J., Drchal, V., Maˇ ca, F., Turek, I. & Khmelevskyi, S. Large anoma- lous Hall angle in the Fe 60Al40 alloy induced by substitutional atomic disorder. Phys. Rev. B101, 054437 (2020)

    2020

  34. [34]

    Mater.23, 821–842 (2011)

    Gutfleisch, O.et al.Magnetic materials and devices for the 21st century: Stronger, lighter, and more energy efficient.Adv. Mater.23, 821–842 (2011)

    2011

  35. [35]

    & Akai, H

    Miyake, T., Harashima, Y., Fukazawa, T. & Akai, H. Understanding and opti- mization of hard magnetic compounds from first principles.Sci. Technol. Adv. Mater.22, 543–556 (2021)

    2021

  36. [36]

    & Becker, J

    Strnat, K., Hoffer, G., Olson, J., Ostertag, W. & Becker, J. J. A family of new cobalt-base permanent magnet materials.J. Appl. Phys.38, 1001–1002 (1967)

    1967

  37. [37]

    & Matsuura, Y

    Sagawa, M., Fujimura, S., Togawa, N., Yamamoto, H. & Matsuura, Y. New material for permanent magnets on a base of Nd and Fe.J. Appl. Phys.55, 2083–2087 (1984)

    2083

  38. [38]

    Herbst, J. F. R 2Fe14B materials: Intrinsic properties and technological aspects. Rev. Mod. Phys.63, 819–898 (1991)

    1991

  39. [39]

    & Izumi, F

    Momma, K. & Izumi, F. VESTA 3 for three-dimensional visualization of crystal, volumetric and morphology data.J. Appl. Crystallogr.44, 1272–1276 (2011)

    2011

  40. [40]

    Slater, J. C. Electronic structure of alloys.J. Appl. Phys.8, 385–390 (1937)

    1937

  41. [41]

    The nature of the interatomic forces in metals.Phys

    Pauling, L. The nature of the interatomic forces in metals.Phys. Rev.54, 899–904 (1938)

    1938

  42. [42]

    Kunimatsu, K.et al.Structure and magnetism in metastablebccCo 1−xMnx epitaxial films.J. Magn. Magn. Mater.548, 168841 (2022)

    2022

  43. [43]

    Patrick, C. E. & Staunton, J. B. Temperature-dependent magnetocrystalline anisotropy of rare earth/transition metal permanent magnets from first principles: 22 the light RCo5 (r= Y, La–Gd) intermetallics.Phys. Rev. Mater.3, 101401 (2019)

    2019

  44. [44]

    Daalderop, G. H. O., Kelly, P. J. & Schuurmans, M. F. H. Magnetocrystalline anisotropy of YCo 5 and related RECo 5 compounds.Phys. Rev. B53, 14415– 14433 (1996)

    1996

  45. [45]

    & Ogura, M

    Okumura, H., Fukushima, T., Akai, H. & Ogura, M. First-principles calculation of magnetocrystalline anisotropy of Y(Co,Fe,Ni,Cu)5 based on full-potential KKR Green’s function method.Solid State Commun.373–374, 115257 (2023)

    2023

  46. [46]

    & Nishio-Hamane, D

    Saito, T. & Nishio-Hamane, D. Synthesis of SmFe 5 intermetallic compound.AIP Adv.10, 015311 (2020)

    2020

  47. [47]

    Buschow, K. H. J. Intermetallic compounds of rare-earth and 3d transition metals. Rep. Prog. Phys.40, 1179–1256 (1977)

    1977

  48. [48]

    Galanakis, I., Dederichs, P. H. & Papanikolaou, N. Origin and properties of the gap in the half-ferromagnetic Heusler alloys.Phys. Rev. B66, 134428 (2002)

    2002

  49. [49]

    I., Irkhin, V

    Katsnelson, M. I., Irkhin, V. Y., Chioncel, L., Lichtenstein, A. I. & de Groot, R. A. Half-metallic ferromagnets: From band structure to many-body effects. Rev. Mod. Phys.80, 315–378 (2008)

    2008

  50. [50]

    A., Mueller, F

    de Groot, R. A., Mueller, F. M., van Engen, P. G. & Buschow, K. H. J. New class of materials: Half-metallic ferromagnets.Phys. Rev. Lett.50, 2024–2027 (1983)

    2024

  51. [51]

    ¨Ozdemir, E. G. & Merdan, Z. Theoretical calculations on half-metallic results properties of FeZrx(x= P, As, Sb and Bi) half-Heusler compounds: density functional theory.Mater. Res. Express6, 086102 (2019)

    2019

  52. [52]

    & Grimme, S

    Friede, M., H¨ olzer, C., Ehlert, S. & Grimme, S. dxtb—An efficient and fully differentiable framework for extended tight-binding.J. Chem. Phys.161, 062501 (2024)

    2024

  53. [53]

    D., Nicholson, D

    Johnson, D. D., Nicholson, D. M., Pinski, F. J., Gyorffy, B. L. & Stocks, G. M. Density-functional theory for random alloys: Total energy within the Coherent- Potential Approximation.Phys. Rev. Lett.56, 2088–2091 (1986)

    2088

  54. [54]

    I., Katsnelson, M

    Liechtenstein, A. I., Katsnelson, M. I., Antropov, V. P. & Gubanov, V. A. Local spin density functional approach to the theory of exchange interactions in ferromagnetic metals and alloys.J. Magn. Magn. Mater.67, 65–74 (1987)

    1987

  55. [55]

    L., Pindor, A

    Gyorffy, B. L., Pindor, A. J., Staunton, J., Stocks, G. M. & Winter, H. A first- principles theory of ferromagnetic phase transitions in metals.J. Phys. F: Met. Phys.15, 1337–1386 (1985)

    1985

  56. [56]

    & Kohn, W

    Hohenberg, P. & Kohn, W. Inhomogeneous electron gas.Phys. Rev.136, B864– B871 (1964). 23

    1964

  57. [57]

    & Sham, L

    Kohn, W. & Sham, L. J. Self-consistent equations including exchange and correlation effects.Phys. Rev.140, A1133–A1138 (1965)

    1965

  58. [58]

    Andersen, O. K. Linear methods in band theory.Phys. Rev. B12, 3060–3083 (1975)

    1975

  59. [59]

    Faulkner, J. S. & Stocks, G. M. Calculating properties with the coherent-potential approximation.Phys. Rev. B21, 3222–3244 (1980)

    1980

  60. [60]

    L., Janak, J

    Moruzzi, V. L., Janak, J. F. & Williams, A. R.Calculated Electronic Properties of Metals(Pergamon Press, New York, 1978)

    1978

  61. [61]

    Advances in Neural Information Processing Systems 32

    Paszke, A.et al.PyTorch: An imperative style, high-performance deep learning library (2019). Advances in Neural Information Processing Systems 32

    2019

  62. [62]

    Don’t unroll adjoint: Differentiating SSA-form programs (2018)

    Innes, M. Don’t unroll adjoint: Differentiating SSA-form programs (2018). ArXiv:1810.07951

    Pith/arXiv arXiv 2018

  63. [63]

    URL https://github.com/jax-ml/jax

    Bradbury, J.et al.JAX: composable transformations of Python+NumPy programs (2018). URL https://github.com/jax-ml/jax

    2018

  64. [64]

    Reverse accumulation and implicit functions.Optim

    Christianson, B. Reverse accumulation and implicit functions.Optim. Methods Softw.9, 307–322 (1998)

    1998

  65. [65]

    Bai, S., Kolter, J. Z. & Koltun, V. Wallach, H.et al.(eds)Deep equilibrium models. (eds Wallach, H.et al.)Adv. Neural Inf. Process. Syst., Vol. 32 (Curran Associates, Inc., 2019). URL https://proceedings.neurips.cc/paper/2019/hash/ 01386bd6d8e091c2ab4c7c7de644d37b-Abstract.html

    2019

  66. [66]

    & Swinburne, T

    Maliyov, I., Grigorev, P. & Swinburne, T. D. Exploring parameter dependence of atomic minima with implicit differentiation.npj Comput. Mater.11, 22 (2025)

    2025

  67. [67]

    Bottou, L., Curtis, F. E. & Nocedal, J. Optimization methods for large-scale machine learning.SIAM Rev.60, 223–311 (2018)

    2018

  68. [68]

    URL https://github.com/google-deepmind/optax

    Hessel, M.et al.Optax: composable gradient transformation and optimisation, in JAX (2020). URL https://github.com/google-deepmind/optax. 24

    2020