Let CSP Be Your ANCHOR: Adaptive Crystal Search over Frozen Structure Priors

Emma Lei Hovmand; Jonas Elsborg; Melih Kandemir; Arghya Bhowmik

cond-mat.mtrl-sciSep 27, 2026

Let CSP Be Your ANCHOR: Adaptive Crystal Search over Frozen Structure Priors

Authors: Emma Lei Hovmand, Jonas Elsborg, Melih Kandemir, Arghya Bhowmik

Organizations: Department of Energy Conversion and Storage, Technical University of Denmark · Department of Mathematics and Computer Science, University of Southern Denmark

Abstract

De novo crystal generation (DNG) models decide where to search in composition space and how to generate structures with one set of weights. We argue that discovery is better served by separating the two. A crystal structure prediction (CSP) model is a physical prior that should be improved by likelihood training, while rewards, including novelty measured against the search's own history, should act on a search over compositions. We introduce ANCHOR, a GRPO composition policy trained with multi-objective rewards around a frozen CSP model, and continuous adaptive novelty (CAN), a graded novelty score against known structures and a growing discovery history. Using the frozen CSP model as a fixed ruler under one evaluator, we test where adaptation should act. Replacing DNG compositions with ANCHOR's policy on the same CSP backbone raises MSUN from 11.4% to 47.6% and SUN from 1.1% to 22.1% at 99.9% formula uniqueness. Fine-tuning DNG models directly on the same rewards instead moves their composition marginal without raising their on-hull fraction. We show that KL-regularized fine-tuning of a DNG model can only reweight chemistry the pretrained model already supports by a bounded factor, while unregularized DNG fine-tunes move toward known or less stable chemistry. Even a stability-only reward routed into ANCHOR's CSP backbone roughly halves SUN relative to the frozen backbone, whereas likelihood training on structures found during search can improve a CSP backbone. Under MatterGen's evaluation pipeline, ANCHOR raises state-of-the-art MSUN from 29.2% to 41.3%, transfers without retraining to two further CSP backbones, and reaches 47.1% after distillation into Crystalite-CSP. As with any model optimised against a potential, its on-hull rate depends on that potential.

Figures & tables

Figure 1 : ANCHOR . A GRPO-trained composition policy queries a frozen crystal diffusion model to generate candidate structures. Candidates are scored by stability, diversity, and continuous adaptive novelty (CAN), a graded RDF-based score that takes the minimum novelty relative to a fixed reference index and the agent’s growing discovery history. Only the composition policy is updated.

Configuration	Valid	Unique	Novel	Meta	Stable	MSUN	SUN	MP-20
Full method	97.5 ±0.4	99.9 ±0.1	97.3 ±0.5	47.7 ±2.0	22.2 ±1.2	47.6 ±2.1	22.1 ±1.2	0 ±0
– count bonus	96.4 ±0.2	100.0 ±0.0	96.4 ±0.3	32.8 ±1.5	12.6 ±0.4	32.8 ±1.5	12.6 ±0.4	0 ±0
– sub-group mask	96.6 ±0.3	99.9 ±0.1	96.5 ±0.1	43.5 ±2.5	11.6 ±0.8	43.4 ±2.4	11.6 ±0.9	0 ±0
– adaptive nov.	97.0 ±0.1	99.8 ±0.1	96.9 ±0.0	39.7 ±2.1	14.7 ±2.0	39.7 ±2.1	14.7 ±2.0	0 ±0
adaptive nov. only	97.7 ±0.6	100.0 ±0.0	97.6 ±0.6	52.0 ±0.4	16.3 ±2.0	52.0 ±0.5	16.3 ±2.0	0 ±0
count bonus only	96.0 ±0.6	100.0 ±0.0	96.0 ±0.6	41.3 ±1.4	12.4 ±0.8	41.3 ±1.4	12.4 ±0.8	0 ±0

Table 1 : Exploration-mechanism ablation (

N_{\text{eval}}=1000

, MatterSim-5M, BFGS-400, positions and lattice free; Appendix E.1 ). Rows give the full method, each mechanism removed (–), each mechanism alone, and the full method with CAN replaced by the binary novelty flag of Chemeleon2 (Appendix F.3 ). Removing adaptive novelty scores novelty against the frozen MP-20 index alone. All columns are percentages except MP-20 , which counts valid samples whose reduced formula occurs in MP-20. Best in MSUN and SUN is bold, second-best underlined. Runs were matched on wall clock. Entries are mean

\pm

std over three sampling seeds.

Untrained composition sources (frozen KLDM-CSP)
Configuration	Valid	Unique	Novel	Meta	Stable	MSUN	SUN	MP-20
Random formulas	9.8 ±0.4	100.0 ±0.0	8.3 ±0.6	2.4 ±0.5	0.4 ±0.1	1.5 ±0.4	0.2 ±0.0	17.5 ±3.5
Enumerated charge-neutral	68.9 ±1.6	100.0 ±0.0	68.2 ±1.3	9.4 ±0.3	1.2 ±0.1	8.8 ±0.0	1.2 ±0.1	9.5 ±4.9
DNG and reward fine-tuned DNG (ours)
Crystalite-DNG	47.0 ±2.7	98.8 ±0.1	31.9 ±1.8	26.2 ±2.1	2.8 ±0.3	14.0 ±0.7	1.5 ±0.4	165.7 ±17.2
$\to$ CSP	48.9 ±2.6	98.8 ±0.1	33.3 ±1.4	28.0 ±3.0	2.9 ±0.6	15.0 ±1.0	1.7 ±0.3	167.3 ±16.6

Table 2 : Separating composition and structure effects. One validity gate, one relaxer (MatterSim-5M, BFGS-400, positions and lattice free), one reference hull, and one fixed MP-20 novelty index for every row;

N_{\mathrm{eval}}=1000

. All columns are percentages except MP-20 , which counts valid samples whose reduced formula occurs in MP-20.

\to

CSP queries the corresponding frozen CSP model with the preceding model’s compositions; FT is DDPO reward fine-tuning at Dirichlet-mean ( mean ) or equal stability and novelty ( bal. ) objective weights. Untrained composition sources are defined in Appendix F.4 . In the likelihood-tuned CSP block, each backbone is fine-tuned by likelihood on MP-20 alone (step-matched control) or with the structures found by ANCHOR added (+ ANCHOR data), and queried with the ANCHOR compositions. Entries are mean ±std over three sampling seeds. Best in MSUN and SUN is bold, second-best underlined. Row definitions in Appendix E.1.1 .

DNG baselines
Model	Unique	Novel	Meta	Stable	$\overline{E}_{h}$	RMSD	MSUN	SUN
MatterGen-MP a	–	–	47.1	–	0.20	0.15	25.8	–
KLDM- $x_{0}$ (D) a	–	–	59.2	–	0.16	0.28	18.5	–
Crystalite b	94.7	56.6	64.5	–	0.15	0.27	24.3	–
MatterGen+REPA c	99.7	63.5	66.2	–	0.12	0.11	29.2	–
Untrained composition sources (frozen KLDM-CSP)

Table 3 : MP-20 under MatterGen’s evaluation pipeline. MSUN is the quantity MatterGen and the cited works report as SUN; SUN applies the same conjunction at

E_{\mathrm{hull}}\leq 0

. All columns are percentages except

\overline{E}_{h}

(eV/atom) and RMSD (Å). Upper block: the strongest published variant of each model family, rounded to one decimal. All published results are in Table 13 . Middle blocks: our own sampling of the released Crystalite-DNG checkpoint and of a KLDM-DNG model we retrained, with their DDPO fine-tunes, which reproduce the published MSUN of their families (

23.6\%

and

18.0\%

against

24.3\%

and

18.5\%

). Our rows report mean ±std over three sampling seeds, with row blocks as in Table 2 . Best in each column is bold, second-best underlined. a Cornet et al. (2025) , b Veljković et al. (2026) , c Zhang et al. (2026) .

Appendix figures & tables21 assets

Supplementary material from the paper’s appendix.

Appendix

Symbol	Type	Meaning
Composition action space
${\mathcal{C}}$	set	discrete chemical space searched by the policy
${\mathcal{E}}$	set	element vocabulary; $E=\|{\mathcal{E}}\|=84$
${\mathcal{K}}$	set	admissible element counts, ${\mathcal{K}}=\{2,3,4\}$ ; $k_{\max}=\max{\mathcal{K}}=4$
$k$	integer	number of distinct elements in an action, $k\in{\mathcal{K}}$
${\bm{e}}$	${\mathcal{E}}^{k}$	the $k$ distinct chosen elements

Appendix

Table 4 : Symbols for the composition policy, reward, and evaluation. Rates are fractions of an evaluation batch of

N_{\mathrm{eval}}=1000

structures, unless stated. Generative-model notation is defined in section 2 .

Block	Elements
Period 1–3	H, Li, Be, B, C, N, O, F, Na, Mg, Al, Si, P, S, Cl
Period 4	K, Ca, Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Cu, Zn, Ga, Ge, As, Se, Br
Period 5	Rb, Sr, Y, Zr, Nb, Mo, Tc, Ru, Rh, Pd, Ag, Cd, In, Sn, Sb, Te, I
Period 6	Cs, Ba, Hf, Ta, W, Re, Os, Ir, Pt, Au, Hg, Tl, Pb, Bi
Lanthanides	La, Ce, Pr, Nd, Pm, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, Lu
Actinides	Ac, Th, Pa, U, Np, Pu

Appendix

Table 5 : The

E=84

elements available to the policy, grouped by block and listed by ascending atomic number within each row. Removed from MP-20: He, Ne, Ar, Kr, Xe.

Symbol / parameter	Value	Role
$r_{\max}$	7 Å	RDF distance cutoff
$n_{\text{bins}}$	80	RDF histogram resolution
$k_{\mathrm{cfg}}$	10	kNN neighbourhood size
$\nu_{0}$	0.75	Bootstrap for first off-MP-20 formula
$C_{\mathrm{hist}}$	20	FIFO cap per reduced formula
$\sigma_{\mathrm{floor}}$	10th pct. calibration	Minimum RBF bandwidth

Appendix

Table 6 : Novelty hyperparameters (main runs).

Parameter	Value	Note
Action space
$E$	84	Element vocabulary (Appendix D.1 )
${\mathcal{K}}$	$\{2,3,4\}$	Distinct elements per action
$N$	20	Max atoms per cell
max atoms / element	12
Policy network

Appendix

Table 7 : Hyperparameters for main runs.

Experiment	Configuration	$G$	Step	Structures
Base-model sweep	$G{=}16$ / 1M	16	7950	$127{,}200$
	$G{=}16$ / 5M	16	7800	$124{,}800$
	$G{=}32$ / 1M	32	3850	$123{,}200$
	$G{=}32$ / 5M	32	3850	$123{,}200$
	$G{=}64$ / 1M	64	2000	$128{,}000$
	$G{=}64$ / 5M	64	1750	$112{,}000$

Appendix

Table 8 : Structures generated by each trained policy,

G\times

(final checkpoint step). All jobs are a single A100 and a seven-day wall. The entropy-schedule horizon is

10^{5}

; this table is what was actually consumed. The selected base model is

G{=}32

/ MatterSim-1M.

Configuration	$N$	Valid	Unique	Novel	Meta	Stable	MSUN	SUN	MP-20
ANCHOR , inference-time backbone
ANCHOR $\to$ KLDM-CSP	1000	97.9	100.0	97.9	38.5	18.1	38.5	18.1	0
ANCHOR $\to$ Crystalite-CSP	1000	94.1	100.0	94.1	37.0	13.7	37.0	13.7	0
ANCHOR $\to$ OMatG-CSP	998	98.2	100.0	98.2	37.9	14.9	37.9	14.9	0
ANCHOR trained with LeMat validity
KLDM backbone	1000	5.6	100.0	5.6	0.7	0.0	0.7	0.0	1

Appendix

Table 9 : Lattice fixed. Internal evaluation pipeline (MatterSim-5M, BFGS-400). Valid = bonds + ICSD oxidation. Unique = distinct formulas /

N

. Novel =

\mathrm{novelty\_score}>0.5

. Stable / Meta:

E_{\mathrm{hull}}\leq 0

0.1

. The last column counts valid samples whose reduced formula occurs in the frozen MP-20 reference index (not ranked). Every ANCHOR row is

0

in the MP-20 column, the DNG rows are not: the novelty reward moves the policy off the reference set, whereas a finetuned generator re-emits it. Table 2 gives the lattice-free values of the headline rows; main-text tables relax the lattice.

Configuration	$N$	Valid	Unique	Novel	Meta	Stable	MSUN	SUN	MP-20
ANCHOR , inference-time backbone
ANCHOR $\to$ KLDM-CSP	1000	72.9	100.0	72.9	25.0	9.4	25.0	9.4	0
ANCHOR $\to$ Crystalite-CSP	1000	72.7	100.0	72.7	26.0	8.4	26.0	8.4	0
ANCHOR $\to$ OMatG-CSP	998	72.8	100.0	72.8	26.2	7.9	26.2	7.9	0
ANCHOR trained with LeMat validity
KLDM backbone	1000	98.3	100.0	97.5	9.2	0.0	8.7	0.0	10

Appendix

Table 10 : Lattice fixed. Internal evaluation pipeline with the LeMat-GenBench validity gate (charge + distance + plausibility) substituted for our own. Relaxer, hull, novelty indices and column definitions are unchanged from Table 9 ; the alternative gate changes which chemistries are admitted rather than acting as a uniformly looser filter. The trained with LeMat validity block was trained against this gate, so its Valid and Novel entries are graded by their own training signal and are not comparable across blocks. Table 2 gives the lattice-free values of the headline rows. Tables in the main text relax the lattice.

		100-step budget		400-step budget
Structures	Relaxation	Conv. (%)	s/struct	Conv. (%)	s/struct
ANCHOR $\to$ KLDM-CSP	positions	16.4	–	98.8	–
	cell free	6.5	7.18	99.4	14.00
ANCHOR $\to$ Crystalite-CSP	positions	27.7	5.47	99.5	8.00
	cell free	9.4	7.08	99.5	12.86
Crystalite-DNG	positions	90.2	2.10	100.0	2.28

Appendix

Table 11 : Relaxation convergence and cost at the training (

100

-step) and evaluation (

400

-step) budgets, on

2{,}820

structures relaxed under both protocols with MatterSim-5M.

100

-step values are read from the

400

-step runs (see text). Training uses positions only at

100

steps; the main-text evaluation frees the lattice at

400

steps. “–”: timing unavailable for that set.

ANCHOR , inference-time backbone
Configuration	$N$	Unique	Novel	Meta	Stable	MSUN	SUN	$\langle E_{\mathrm{h}}\rangle$	RMSD
ANCHOR $\to$ KLDM-CSP	993	100.0	100.0	42.3	17.1	42.3	17.1	0.133	1.40
ANCHOR $\to$ Crystalite-CSP	993	100.0	100.0	42.8	14.5	42.8	14.5	0.138	1.27
ANCHOR $\to$ OMatG-CSP	991	100.0	100.0	43.2	17.3	43.2	17.3	0.134	1.28
ANCHOR trained with LeMat validity
KLDM backbone	998	100.0	97.2	22.0	0.3	19.4	0.0	0.189	0.60

Appendix

Table 12 : MatterGen evaluation pipeline (unmodified), which supplies its own relaxer, reference hull and novelty definition. MSUN is unique

\wedge

novel

\wedge

E_{\mathrm{hull}}\leq 0.1

, which the pipeline reports as S.U.N.; SUN uses

E_{\mathrm{hull}}\leq 0

. Meta is the stability rate at

E_{\mathrm{hull}}\leq 0.1

; Stable is the raw stability rate at

E_{\mathrm{hull}}\leq 0

, unfiltered by uniqueness or novelty.

\langle E_{\mathrm{hull}}\rangle

(eV/atom) and RMSD (Å, against the pre-relaxation cell): lower is better.

N

is the number of structures the pipeline scored, below

1000

where it dropped structures of its own accord. We cannot report RMSD for Crystalite-DNG

\to

CSP for its particular structures: the official pipeline fails while computing it. The other columns on that row use the same relaxed cells. Rows are defined in Appendix E.1 . The pipeline relaxes positions and lattice (FIRE with ExpCellFilter ). Entries are from sampling seed 0, and Table 3 reports the three-seed means.

Published DNG baselines
Model	Unique	Novel	Meta	$\overline{E}_{h}$	RMSD	MSUN
MatterGen-MP a	–	–	47.1	0.20	0.15	25.8
DiffCSP a	–	–	41.3	0.19	0.41	20.1
KLDM- $x_{0}$ (C) a	–	–	38.6	0.27	0.37	16.7
KLDM- $x_{0}$ (C-AB) a	–	–	49.8	0.19	0.30	17.9
KLDM- $x_{0}$ (D) a	–	–	59.2	0.16	0.28	18.5

Appendix

Table 13 : Representative published DNG results on MP-20 under MatterGen’s evaluation pipeline, rounded to one decimal, with the frozen-backbone ANCHOR rows of Table 3 for reference (mean ±std over three sampling seeds). MSUN is the quantity MatterGen and the cited works report as SUN. Table 3 keeps the strongest variant of each model family. Best in each column is bold, second-best underlined. a Cornet et al. (2025) , b Höllmer et al. (2025) , c Veljković et al. (2026) , d Zhang et al. (2026) .

ANCHOR , inference-time backbone
Configuration	$N$	Valid	Unique	Novel	Meta	Stable	$\langle E_{\mathrm{h}}\rangle$	MSUN	SUN
ANCHOR $\to$ KLDM-CSP	1000	73.0	73.0	73.0	2.9	0.0	0.20	2.9	0.0
ANCHOR $\to$ Crystalite-CSP	1000	73.0	73.0	73.0	3.1	0.0	0.20	3.1	0.0
ANCHOR $\to$ OMatG-CSP	998	72.9	72.9	72.9	3.1	0.0	0.20	3.1	0.0
ANCHOR trained with LeMat validity
KLDM backbone	1000	99.4	99.4	96.2	28.1	0.1	0.17	26.0	0.0

Appendix

Table 14 : LeMat-GenBench evaluation against the Bulk ensemble hull, NequIP pre-relaxation unless noted. All rates are fractions of the submitted

N

, with validity, uniqueness and novelty applied in that order.

\langle E_{\mathrm{hull}}\rangle

(eV/atom): lower is better. The trained with LeMat validity block optimised a LeMat validity filter during training, so its Valid, Unique and Novel entries are not comparable across blocks. Rows are defined in Appendix E.1 .

ANCHOR , inference-time backbone
Configuration	$N$	Valid	Unique	Novel	Meta	Stable	$\langle E_{\mathrm{h}}\rangle$	MSUN	SUN
ANCHOR $\to$ KLDM-CSP	1000	73.0	73.0	73.0	8.2	0.4	0.170	7.8	0.4
ANCHOR $\to$ Crystalite-CSP	1000	73.0	73.0	73.0	8.6	0.8	0.172	7.8	0.8
ANCHOR $\to$ OMatG-CSP	998	72.9	72.9	72.9	9.1	0.6	0.170	8.5	0.6
ANCHOR trained with LeMat validity
KLDM backbone	1000	99.4	99.4	98.9	34.0	1.1	0.154	32.5	1.0

Appendix

Table 15 : LeMat-GenBench evaluation against the MP-20 ensemble hull, following the published Table 11 protocol so the rows can be read against that leaderboard. Columns as in Table 14 ; only the reference hull changes. The same caveat applies to the trained with LeMat validity block.

Figure 2 : Training gate sensitivity. Policies trained with KLDM-CSP against our validity gate and against the LeMat-GenBench gate. Left: elements appearing in formulas over the last 400 training steps. Middle: fraction of all-metal compositions during training. Right: mean atoms per formula.

ANCHOR trained against the internal validity gate
	Internal gate			LeMat gate, our pipeline			LeMat-GenBench
Configuration	Valid	SUN	MSUN	Valid	SUN	MSUN	Valid	SUN	MSUN
ANCHOR $\rightarrow$ KLDM-CSP	97.9	23.2	49.9	72.9	13.6	34.1	73.0	0.4	7.8
ANCHOR trained against the LeMat validity gate
KLDM backbone	5.6	0.0	2.1	98.3	0.1	28.1	99.4	1.0	32.5
DNG reference

Appendix

Table 16 : The gate and the threshold decide the ranking. Identical structures scored three ways. Internal and LeMat gate differ only in the validity definition; relaxer (MatterSim-5M, BFGS-400, lattice free), hull and novelty index are held fixed (Table 2 ). LeMat-GenBench is that benchmark’s own pipeline against its MP-20 ensemble hull, so its columns follow the published leaderboard. SUN is on-hull throughout, MSUN is

E_{\mathrm{hull}}\leq 0.1

. Single sampling seed (seed 0), so the Internal-gate columns differ slightly from the three-seed means of Table 2 . Policies in the second block were trained against the LeMat validity filter.

Rank	Config	Valid%	Unique%	SUN%	MSUN%
1	G32 / 1M	97.9	100.0	18.1	38.5
2	G16 / 5M	94.3	100.0	8.6	20.8
3	G16 / 1M	93.6	100.0	6.0	13.5
4	G64 / 1M	96.5	100.0	3.5	13.4
5	G32 / 5M	96.0	100.0	3.9	11.5
6	G64 / 5M	95.4	100.0	1.5	8.5

Appendix

Table 17 : Lattice fixed. Post-training evaluation. Unique% is computed over all structures.

\mathrm{SUN}

E_{\mathrm{hull}}\leq 0

;

\mathrm{MSUN}

E_{\mathrm{hull}}\leq 0.1\,\mathrm{eV/atom}

Rank	Config	Valid%	Unique%	SUN%	MSUN%
1	G32 / 1M	87.0	99.8	7.2	22.1
2	G16 / 5M	88.0	99.7	3.8	11.7
3	G16 / 1M	85.1	99.8	3.3	11.2
4	G64 / 5M	85.2	99.8	1.2	7.5
5	G32 / 5M	84.8	99.8	2.1	7.5
6	G64 / 1M	84.9	99.8	1.4	7.0

Appendix

Table 18 : Late-window comparison on the final

10\mathrm{k}

training structures. Unique% is computed over all structures.

\mathrm{SUN}

\mathrm{MSUN}

use training-time MatterSim scores (stability

{=}1.0

{\geq}0.9

as proxies for

E_{\mathrm{hull}}\leq 0

{\leq}0.1

$r_{\mathrm{inv}}$	Valid%	Bond%	Charge%	Unique%	SUN%	MSUN%
$-0.1$	96.6	96.7	99.9	100.0	8.7	22.0
$-0.2$ (selected)	97.9	98.4	99.5	100.0	18.1	38.5
$-0.5$	97.4	97.6	99.8	100.0	12.0	28.4

Appendix

Table 19 : Lattice fixed. Invalid-penalty sweep on the selected base model (

G{=}32

, MatterSim-1M), evaluated post-training under Appendix E.1 (

N{=}1000

, seed 0). Bond and Charge are the pass rates of the two components of the validity gate (Appendix D.2 ); Valid is the conjunction.

Configuration	Adapt.	CB	SG	Valid (%)	Unique (%)	Stable (%)	Meta (%)	MSUN (%)	SUN (%)
Full method	$\bullet$	$\bullet$	$\bullet$	97.9	100.0	18.1	38.5	38.5	18.1
– count bonus	$\bullet$	–	$\bullet$	96.3	100.0	8.7	24.6	24.6	8.7
– sub-group mask	$\bullet$	$\bullet$	–	96.4	100.0	7.6	27.9	27.9	7.6
– adaptive nov.	–	$\bullet$	$\bullet$	96.9	99.9	8.4	26.4	26.4	8.4
adaptive nov. only	$\bullet$	–	–	98.1	100.0	12.3	37.2	37.1	12.3
count bonus only	–	$\bullet$	–	96.4	100.0	7.3	28.3	28.3	7.3

Appendix

Table 20 : Lattice fixed. Exploration mechanism ablation, evaluated post-training (

N_{\text{eval}}=1000

, full chemical space, shared MatterSim-5M evaluation potential, BFGS relaxation of positions only for at most 400 steps, see Appendix E.1 ).

\bullet

marks that the mechanism is enabled, – means disabled. Adapt. = novelty scored against the growing agent-history index or frozen MP-20 index; bin. = the binary Chemeleon2 flag of Appendix F.3 . CB = count bonus, SG = element sub-group mask. Runs were matched on wall clock.

Composition source	Valid%	Unique%	SUN%	MSUN%
KLDM + random formulas	9.5	100.0	0.2	0.4
KLDM + enumerated charge-neutral	70.1	100.0	0.8	3.1
ANCHOR (selected policy)	97.9	100.0	18.1	38.5

Appendix

Table 21 : Lattice fixed. Effect of composition selection with the structural pipeline held fixed. All rows use the same frozen KLDM-CSP backbone and post-training evaluation protocol (

N{=}1000

, seed 0, full chemical space, shared MatterSim-5M, BFGS

{=}400

). Unique% is distinct reduced formulas over all

N

	Crystalite-DNG	KLDM-DNG
Initialization	released checkpoint	MP-20 retrain (below)
Learning rate	$5\times 10^{-6}$	$5\times 10^{-6}$
Batch size (structures / episode)	16	16
Denoising steps per sample	150	50
Scored steps ${\mathcal{T}}$	churn $\in[0.2,2.0]$	all
Inner epochs per batch	1	1

Appendix

Table 22: DDPO fine-tuning of the two DNG models. One Adam step per episode, taken after accumulating gradients over the scored denoising steps of the batch.

Configuration	MSUN	SUN
KLDM-CSP
Released checkpoint	42.3	17.1
FT: MP-20 only (ctl)	41.9	17.1
FT: MP-20 $+$ ANCHOR	41.6	16.5
FT: MP-20 $+$ ANCHOR , unrelaxed cells	42.5	16.7
FT: MP-20 $+$ Crystalite-rendered ANCHOR	43.9	18.9

Appendix

Table 23 : Training the CSP backbone on ANCHOR structures, under the MatterGen evaluation protocol. ANCHOR contributed

27{,}136

structures from one of its training runs; each

+

ANCHOR arm adds them to MP-20 and each ctl arm is the step-matched control on pristine MP-20, so the pair isolates the corpus from the extra optimisation. Finetuning resumes from the released checkpoint; from scratch retrains it for the same

699.6

k (KLDM) or

1.4

M (Crystalite) steps. Every row is scored on the same

1000

ANCHOR compositions as the inference-time backbone rows of Appendix E.1 ; only the CSP checkpoint changes. Entries are from sampling seed 0; Tables 2 and 3 report seed-averaged values for the FT arms. Composition validity and uniqueness are fixed by the shared formula list and are reported per protocol in Appendix E.1 .

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Sep 22, 2026cond-mat.mtrl-sci

Deep Generative Crystal Structure Prediction: A Benchmark Study and a Controlled Test of Prototype Dependence

Deep generative models are widely reported to enable de novo crystal structure prediction (CSP), but their capability has not been measured consistently against template-based methods. We evaluate 12 representative generative CSP models, spanning latent-variable, diffusion, flow-matching, autoregressive, and manifold random-walk architectures, against TCSP 2.0 on 180 test structures and a leakage-controlled subset of 46. All methods use identical structure-matching, symmetry, and consensus criteria. Template retrieval is the strongest single method, reaching 68.3% top-1 success; symmetry-aware EquiCSP (66.4%) and Uni-3DAR (62.9%) form the next tier. However, comparison with TCSP 2.0 shows that most structures correctly predicted by generative models are also correctly predicted by template substitution. Thus, the set of structures uniquely reachable by generation is small, limiting its practical advantage for discovering structures outside existing prototype libraries. To test the source of this performance, we removed entire stoichiometric prototype families from the training set and retrained the strongest generative model. Accuracy declined by 50-78% across four families, establishing that performance is substantially prototype-dependent. A small minority of structures survived removal of their prototype family, demonstrating a real but limited retrieval-independent predictive capacity. Present generative CSP models therefore function largely as implicit, softer-edged prototype libraries rather than genuinely de novo predictors. Enlarging this residual capacity, rather than aggregate match rate alone, is the central open problem.

Lai Wei, Rongzhi Dong, Ying Feng +2

Department of Computer Science and Engineering University of South Carolina Columbia, SC 29201 · Department of Computer Science and Cybersecurity, University of North Georgia, Dahlonega, GA 30597

Crystal Structure Prediction Molecular Generation Generative Models Symmetry

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Sep 9, 2026cond-mat.mtrl-sci

uFlowCSP: Crystal Structure Prediction using Mean flow generative models

Crystal structure prediction (CSP) is fundamental to computational materials discovery. Generative models including CDVAE, DiffCSP, FlowMM, and CrystalFlow learn stable-crystal distributions directly, but diffusion and flow-matching inference requires tens to thousands of sequential network evaluations per candidate. We introduce uFlowCSP, a MeanFlow-based CSP model that learns the average, rather than instantaneous, probability-flow velocity. It generates a complete structure in one to five evaluations, delivering 5x-58x faster inference with equal or better performance. A chemistry- and symmetry-aware Transformer uses canonical atom ordering, global composition, and per-token chemistry embeddings. A coarse crystal-system token is used only during training; it provides additive gains, particularly improving space-group agreement despite being absent at inference, which remains formula-only. On MP-20 with 20 candidates per target, one step matches CrystalFlow (78.38% vs. 78.34%) with 100x fewer evaluations and about 10x lower wall-clock time. Five steps reach 83.64%, exceeding CrystalFlow (78.34% at 2,000 evaluations) and DiffCSP (77.93% at about 20,000), while using 20x fewer evaluations. uFlowCSP generates 10,000 structures in 0.39-1.31 minutes, versus 6.5 for CrystalFlow and 76.1 for DiffCSP. Under CSPBench's energy-ranked top-five structure-and-space-group criterion, five-step uFlowCSP reaches 72%/72%/65% structure, space-group, and consensus match rates. CrystalFlow reaches 78%/73%/68% at 100 steps but falls to 49%/32%/31% at five. Thus, uFlowCSP improves accuracy per network evaluation, not merely peak accuracy.

Sourin Dey, Dipannoy Das Gupta, Lai Wei +2

Department of Computer Science and Engineering University of South Carolina Columbia, SC 29201 · Department of Computer Science and Cybersecurity University of North Georgia Columbia, SC 29201

Crystal Structure Prediction Crystal Structure Meanflow Generative Models Token Embeddings

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Aug 6, 2026cs.LG

CrystalGRPO: Target-Aligned and Coverage-Preserving Reinforcement Learning for Flow-Based Crystal Structure Prediction

Flow-based generative models can efficiently produce candidate structures for crystal structure prediction (CSP), but their pretrained objectives do not directly optimize downstream target recovery. Reinforcement-learning post-training offers a flexible solution, yet existing approaches rely primarily on energy rewards and coordinate-only stochastic policies. Predicted energy does not identify the reference polymorph, while reward-driven concentration can reduce the candidate coverage required for Top-N recovery. We introduce CrystalGRPO, a CSP-aligned post-training framework that extends existing ODE-to-SDE policy constructions to the joint coordinate--lattice state. CrystalGRPO combines MACE-predicted energy with a StructureMatcher-based recovery score and provides two operating modes: CrystalGRPO-Q, which prioritizes single-draw recovery, and CrystalGRPO-C, which combines full-trajectory reference regularization with a coverage-aware group advantage to preserve finite-budget target recovery. Across MP-20 and MPTS-52 with PXRDGen and OMatG backbones, both variants reduce one- and twenty-sample RMSE relative to coordinate-only reinforcement in all four backbone--dataset settings. CrystalGRPO-Q consistently improves Top-1, whereas CrystalGRPO-C achieves a higher Top-20 across all settings.

Kaixiang Su, Hongfei Xue, Qiang Zhu

Department of Computer Science, University of North Carolina at Charlotte · Department of Mechanical Engineering and Engineering Science, University of North Carolina at Charlotte · North Carolina Battery Complexity, Autonomous Vehicle and Electrification (BATT CAVE) Research Center

Crystal Structure Prediction Crystal Generative Flow Networks Post-Training

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