Deep Scientific Reasoning under Physical Constraints: Structure-Aware Spectrum Prediction
Authors: Yingheng Wang, Tao Yu, Shufeng Kong, Francesco Ricci, John M. Gregoire, Carla P. Gomes
Organizations: Department of Computer Science, Cornell University · Griffith University · Institute of Condensed Matter and Nanosciences, Universit´e catholique de Louvain · Matgenix SRL · California Institute of Technology · Lila Sciences
Structured scientific spectra encode rich physical information while obeying hard constraints, such as conservation laws and characteristic spectral geometry. Accurate prediction of these spectra is central to materials discovery, yet existing methods often treat them as unconstrained sequences and therefore fail to enforce the underlying physical structure. Taking electronic density of states (eDOS) as the prototypical example, we introduce DeepSciReasoner, a general paradigm for predicting scientific spectra under physical constraints. The framework combines structure-aware spectrum decoding with constraint-preserving physical reasoning, allowing predictions to capture rich spectral structure while respecting the underlying physics. We evaluate DeepSciReasoner on eDOS, phonon density of states (phDOS), X-ray absorption near-edge structure (XANES), and Raman spectra, where it substantially improves prediction accuracy while maintaining physical consistency. These results establish DeepSciReasoner as a reusable blueprint for structured scientific spectrum prediction under hard physical constraints.
Figures & tables
Figure 1: DeepSciReasoner consists of two stages, illustrated here with eDOS as the running example. ( Left ) Structure-Aware Spectrum Decoding: the input crystal structure is encoded by an edge-aware crystal graph Transformer to produce atom-level embeddings, which are decoded into an initial spectrum via source (atom-to-spectral-axis) attention over discrete bins, preserving atom-projected contributions. ( Right ) Constraint-Preserving Physical Reasoning: the initial normalized spectrum is refined through a learned velocity field that transports spectral mass along the axis while conserving total mass, yielding progressively sharper band edges and peaks and producing the final prediction.
Figure 2: Structure-aware Spectrum decoding. The edge-aware crystal graph Transformer encodes the input structure and a decoder with hierarchical attention predicts the eDOS, where each energy bin selectively aggregates atom-projected contributions.
CalTech
TRI
Method
R 2↑
MAE ↓
MSE ↓
WD ↓
R 2↑
MAE ↓
MSE ↓
WD ↓
E3NN
0.748 ± 0.002
2.421 ± 0.003
27.351 ± 0.629
0.261 ± 0.005
0.669 ± 0.004
2.945 ± 0.005
36.535 ± 0.532
0.445 ± 0.004
GATGNN
0.772 ± 0.001
2.272 ± 0.016
24.831 ± 0.585
0.230 ± 0.004
0.702 ± 0.004
2.724 ± 0.004
33.811 ± 0.420
0.370 ± 0.003
CrystalFormer
0.765 ± 0.006
2.344 ± 0.016
26.138 ± 0.477
0.248 ± 0.012
0.695 ± 0.010
2.869 ± 0.082
34.584 ± 0.581
0.399 ± 0.017
Mat2Spec
0.743 ± 0.002
2.226 ± 0.010
27.955 ± 0.137
0.228 ± 0.013
0.677 ± 0.002
2.719 ± 0.072
36.628 ± 0.105
0.395 ± 0.009
DOSTransformer
0.771 ± 0.002
2.023 ± 0.032
24.675 ± 0.354
0.212 ± 0.040
0.716 ± 0.003
2.455 ± 0.080
32.208 ± 0.288
0.375 ± 0.014
Table 1: Performance of eDOS prediction on two benchmark datasets. We report mean ± one standard deviation. Darkly shaded cells indicate the best results and lightly shaded cells indicate the second best.
XANES-full
XANES-O
Raman
phDOS
Method
R 2↑
WD ↓
R 2↑
WD ↓
R 2↑
WD ↓
R 2↑
MAE ↓
MSE ↓
WD ↓
E3NN
0.650 ± 0.012
0.108 ± 0.003
0.381 ± 0.011
0.099 ± 0.004
0.096 ± 0.005
0.451 ± 0.019
0.506 ± 0.015
0.082 ± 0.003
0.026 ± 0.001
0.103 ± 0.004
GATGNN
0.648 ± 0.005
0.104 ± 0.001
0.293 ± 0.013
0.108 ± 0.002
0.115 ± 0.005
0.526 ± 0.008
0.270 ± 0.025
0.106 ± 0.002
0.038 ± 0.003
0.155 ± 0.008
MEGNet
0.657 ± 0.008
0.107 ± 0.002
0.347 ± 0.020
0.102 ± 0.006
0.045 ± 0.004
0.449 ± 0.024
0.538 ± 0.014
0.080 ± 0.003
0.024 ± 0.001
0.094 ± 0.004
M3GNet
0.648 ± 0.010
0.107 ± 0.002
0.386 ± 0.012
0.096 ± 0.005
0.083 ± 0.003
0.426 ± 0.021
0.543 ± 0.012
0.075 ± 0.003
0.023 ± 0.001
0.075 ± 0.003
DSR
0.764 ± 0.009
0.098 ± 0.003
0.474 ± 0.010
0.093 ± 0.003
0.264 ± 0.002
0.423 ± 0.017
0.636 ± 0.013
0.062 ± 0.004
0.021 ± 0.002
0.061 ± 0.002
Table 2: Generalization to other structured spectra. Performance on XANES-full (full multi-absorber), XANES-O (oxygen K-edge subset), Raman, and phDOS benchmarks. Darkly shaded cells denote the best results and lightly shaded cells denote the second best. Abbreviations: DSR = DeepSciReasoner.
CalTech
TRI
Method
Prec. ↑
Recall ↑
F1 ↑
MCC ↑
Prec. ↑
Recall ↑
F1 ↑
MCC ↑
E3NN
0.672 ± 0.035
0.366 ± 0.013
0.474 ± 0.003
0.479 ± 0.011
0.647 ± 0.052
0.314 ± 0.071
0.423 ± 0.028
0.442 ± 0.012
GATGNN
0.699 ± 0.033
0.430 ± 0.008
0.532 ± 0.003
0.532 ± 0.009
0.671 ± 0.049
0.362 ± 0.053
0.476 ± 0.021
0.489 ± 0.008
CrystalFormer
0.654 ± 0.023
0.443 ± 0.031
0.532 ± 0.022
0.542 ± 0.013
0.616 ± 0.037
0.371 ± 0.009
0.465 ± 0.024
0.472 ± 0.031
Mat2Spec
0.726 ± 0.018
0.551 ± 0.007
0.629 ± 0.004
0.611 ± 0.019
0.601 ± 0.024
0.429 ± 0.016
0.509 ± 0.017
0.508 ± 0.014
DOSTransformer
0.684 ± 0.027
0.532 ± 0.009
0.594 ± 0.014
0.584 ± 0.051
0.573 ± 0.073
0.329 ± 0.020
0.416 ± 0.052
0.422 ± 0.066
Table 3: VB-gap identification performance. We report mean ± one standard deviation. Darkly shaded cells denote the best results and lightly shaded cells denote the second best among applicable methods.
CalTech
TRI
Method
R 2↑
MAE ↓
MSE ↓
WD ↓
R 2↑
MAE ↓
MSE ↓
WD ↓
DSR (full)
0.815 ± 0.007
1.983 ± 0.021
19.612 ± 0.225
0.191 ± 0.017
0.761 ± 0.009
2.357 ± 0.046
27.162 ± 0.563
0.305 ± 0.011
DSR-NR
0.807 ± 0.003
2.061 ± 0.020
20.990 ± 0.373
0.207 ± 0.005
0.742 ± 0.005
2.420 ± 0.018
29.387 ± 0.641
0.329 ± 0.004
DSR-NDNR
0.779 ± 0.001
2.259 ± 0.003
24.726 ± 0.174
0.223 ± 0.007
0.708 ± 0.003
2.701 ± 0.005
33.196 ± 0.136
0.372 ± 0.003
Table 4: Ablation study of DeepSciReasoner on eDOS prediction. We report mean ± one standard deviation. Darkly shaded cells denote the best results and lightly shaded cells denote the second best. Abbreviations: DSR = DeepSciReasoner; NR = w/o constraint-preserving physical reasoning; NDNR = w/o both components.
Figure 3: Data scaling behavior on the CalTech dataset. Performance of different methods as a function of training set size (25%, 50%, 75%) on R 2 , MAE, MSE, and WD.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Component / Setting
Value
Edge-Aware Crystal Graph Transformer (Encoder)
Backbone operator
TransformerConv ( Fey & Lenssen, 2019 )
# encoder layers
3
Hidden size
512
Node embedding dim dh
512
Edge embedding dim de
512
Appendix
Table 5: Hyperparameter settings for DeepSciReasoner.
Dataset
Metric
CalTech
TRI
Hardware Specifications
GPU Specification
1 × NVIDIA H100 (80GB)
CPU Specification
Intel Xeon Platinum 8462Y+
Stage I: Encoder + Decoder Training
Encoding Time / Crystal (ms)
4.39
4.75
Appendix
Table 6: Computational cost and hardware specifications on CalTech and TRI datasets.
Figure 4: Data scaling behavior on the TRI dataset. Performance of different methods as a function of training set size (25%, 50%, 75%) on R 2 , MAE, MSE, and WD.
Figure 5: Representative examples from the CalTech (Top) and TRI (Bottom) test sets compare predicted eDOS of each method (including DeepSciReasoner and the baselines) against DFT-derived targets.
Figure 6: Representative examples from the CalTech (Top) and TRI (Bottom) test sets compare predicted eDOS of DeepSciReasoner against DFT-derived targets.
State Key Laboratory of AI Safety, Institute of Computing Technology, Chinese Academy of Sciences, China · University of Chinese Academy of Sciences, China