SKG in Practice: OOD Generalization Cases Across FourDomains

Jian An; Chaoshi Wu; Xinpan Luo

cs.LGOct 23, 2026

SKG in Practice: OOD Generalization Cases Across FourDomains

Authors: Jian An, Chaoshi Wu, Xinpan Luo

Abstract

This paper presents the engineering implementation methods of the SKG inference framework and provides complete case studies across four domains. Case 1, 2D Dissipative Mechanical System: OOD MSE is 1.69 to 4.12 times lower than the best neural baseline; the state-dependent metric yields a 10^8-order improvement under controlled conditions. Case 2, Target Detection Sim-to-Real: Real-machine mAP increases from 0.2564 to 0.4964 (+24.00 points, single-run experiment, 32 images, point estimate), and AP75 increases from 0.1736 to 0.5765 (+40.29 points), with no real-machine retraining required. Case 3, Sequence Modeling sine to square: OOD MSE is 0.0296 versus Transformer 0.0641 (mean of 5 seeds, p=0.024). Case 4, Molecular Dynamics MD17: ID@100 ADE is 0.0285 versus ESTAG 0.1467; OOD@100 is 0.0316 versus 0.1411 (single seed). For each case, we present the objective, applied SKG theory, parameter settings, baseline comparison protocol, data results, and sources of advantage. Following the cases, we provide the complete methodology: pre-validation, task invariant construction, three-stage training, residual training, and the four theorems and one proposition supporting these methods. This paper not only demonstrates effective configurations of SKG across four domains but also honestly reports the variance of single-run experiments, limitations of single-seed results, negative updates caused by parameterization mismatch, and practical experience of using engineering thresholds as diagnostic signals rather than hard constraints.

Figures & tables

Case	Domain	Key Results
1	2D Dissipative Mechanical System	OOD MSE is 1.69–4.12 times lower than the best neural baseline; state-dependent metric yields a $10^{8}$ -order improvement under controlled conditions
2	Target Detection Sim-to-Real	Real-machine mAP 0.2564 $\to$ 0.4964 (+24.00 points, single-run experiment, 32 images, point estimate), AP75 0.1736 $\to$ 0.5765 (+40.29 points), no real-machine retraining required
3	Sequence Modeling sine $\to$ square	OOD MSE 0.0296 vs Transformer 0.0641 (mean of 5 seeds, p=0.024)
4	Molecular Dynamics MD17	ID@100 ADE 0.0285 vs ESTAG 0.1467; OOD@100 0.0316 vs 0.1411 (single seed)

Table 1

Component	Configuration
Symplectic structure ( $\omega$ )	Constant matrix
Mass ( $M_{m}$ )	Unknown constant positive diagonal, calibrated from training data
Potential energy ( $V\subset H$ )	Gaussian kernel Nyström, 96 centers
Dissipation ( $g$ )	Full symmetric matrix field, Gaussian kernel Nyström, softplus
Training	Weak-form residual, LBFGS, 500 iterations
Regularization	Ridge $10^{-6}$ + Nyström spectral truncation

Table 2

Component	Configuration
Detector	Co-DETR (ResNet-50), with supervised angle branch and rotated width/height branch
Detector training	Fine-tuned only on the simulation training set (50-epoch budget, taking epoch 48), frozen and directly used for real-machine test
Local coordinates	$u,v$ normalized by detector angle rotation; core region $r\leq 0.6$ , peripheral region $r>1.05$
Contour extraction	Multi-channel local Otsu (grayscale/saturation/value) + RGB GMM-GrabCut fallback
State	$q_{rel}=(\frac{c_{x}-c_{0x}}{L},\frac{c_{y}-c_{0y}}{L},\log\frac{w}{L},\log\frac{h}{L},\theta)$
Energy ( $H,S$ )	Conditional target energy, angular wrapped difference

Table 7

Data Domain	Method	OBB mAP	AP50	AP75
Simulation Dev	Co-DETR Baseline	0.6469	0.8914	0.7195
	+ SKG 5D Flow	0.7687 (+12.18)	0.9600	0.8410 (+12.15)
Real-Machine Test	Co-DETR Baseline	0.2564	0.6730	0.1736
	+ SKG 5D Flow	0.4964 (+24.00)	0.8432	0.5765 (+40.29)

Table 8

Component	Form	Parameter Count
State	Encoder MLP(24 $\to$ 64 $\to$ 16)	$\sim$ 1.7K
Metric ( $g$ )	$G(z)=\operatorname{diag}(d)+UU^{\top}$ , rank 4	$\sim$ 1.8K
Energy ( $H,S$ )	quad + bound + mlp + spec	$\sim$ 5.5K
Step size	$\eta(z)=\exp(\log\eta_{0}+\mathrm{MLP}(z))$	$\sim$ 500
Evolution	10-step explicit evolution + 3 backtracking	—
Decoder	Linear layer $16\to 1$	17

Table 14

Stage	Training Object	Loss	Epoch
A	encoder + metric + energy + decoder + $\eta$	Trajectory loss	100
B	metric + energy + $\eta$	Weak-form residual (Simpson)	100
C	decoder	Trajectory loss	50

Table 15

Metric	SKG	Transformer
ID MSE	0.003023 $\pm$ 0.000223	0.003099 $\pm$ 0.000296
OOD MSE	0.029577 $\pm$ 0.008228	0.064096 $\pm$ 0.018351
OOD/ID	9.953 $\pm$ 3.251	21.036 $\pm$ 6.893

Table 16

Component	Configuration
Baseline	MACE-OFF23 medium (frozen)
Residual network	Conservative Residual EGNN
Hidden dimension	64
Layers	3
RBF count	32
Cutoff radius	5.0 Å

Table 20

Method	Structure	Parameters	Training Objective
ESTAG-res	Temporal graph network + equivariant message passing	478,622	$\\|\Delta R_{\theta}-(R_{\text{true}}-R_{\text{MACE}})\\|^{2}$
ATOM-res	Global neural operator	$\sim$ 1M	Same as above
GSE-res	Velocity field integration	—	Same as above

Table 21

Baseline	Parameters	Source
ESTAG [ 3 ]	7,799	NeurIPS 2023
GSE-Flow [ 4 ]	25,208	ICML 2026
GF-NODE [ 5 ]	531,986	TMLR 2025
ATOM [ 6 ]	3,557,830	ICLR 2026
MACE-Cart	0 (frozen)	Reference baseline

Table 22

Model	Parameters	@10	@50	@100
SKG Potential Residual	$\sim$ 3K	0.003853	0.017989	0.028546
ESTAG	7,799	0.006718	0.065849	0.146704
MACE-Cart	0	0.006710	—	—
GF-NODE	531,986	0.037661	0.145711	0.219097
GSE-Flow	25,208	0.037954	0.223124	2.761314
ATOM	3,557,830	0.068298	0.718328	1.152361

Table 23

Model	@10	@50	@100
SKG Potential Residual	0.004144	0.018740	0.031575
ESTAG	0.006977	0.063881	0.141094

Table 24

Method	Parameters	@10	@50	@100	Relative to Potential @10	@100
Potential Residual	$\sim$ 3K	0.003853	0.017989	0.028546	—	—
ESTAG-res	478,622	0.004567	0.024961	0.040293	+18.5%	+41.2%
ATOM-res	$\sim$ 1M	0.004591	0.025167	0.040385	+19.2%	+41.5%
MACE-Cart	0	0.006706	0.028402	0.043124	+74.0%	+51.0%

Table 27

Method	@10	@50	@100
Potential Residual	0.004144	0.018740	0.031575
ESTAG-res	0.005046	0.025979	0.042990
ATOM-res	0.005502	0.026796	0.043456
MACE-Cart	0.006763	0.029043	0.045503

Table 28

Dimension	Case 1	Case 2	Case 3	Case 4
Domain	Scientific ML	Industrial Vision Sim-to-Real	Sequence Modeling	Molecular Dynamics
Dynamics	Second-order dissipative	First-order dissipative	First-order gradient flow	Second-order Newtonian
Paradigm	Three-stage A/B/C	Unit metric + energy alignment	Three-stage A/B/C	Known metric (mass matrix)
Evidence Strength	Relatively Strong	Moderate	Weak	Moderate
Pre-validation	Three criteria passed	Physical contour driven	Passed	Implicit
Task Invariant ( $\Delta$ )	Gaussian kernel potential	Boundary energy (hierarchical contour)	Spectral prior	E(3) equivariant

Table 31

Criterion	Test	Passing Standard
Truly invariant	Apply 24 transformations to the OOD transformation group, compute relative change rate	mean < 0.05, max < 0.10
Non-trivial	Variance of each dimension	> $10^{-6}$
Non-redundant	Cov condition number, inter-dimensional correlation	Condition number < $10^{3}$ , correlation < 0.95

Table 32

Criterion	Test	Passing Standard
Condition number	$\kappa(g_{z})$	< $10^{3}$
Directional stability	Direction cosine when metric changes	> 0.95
Full rank	$J_{z}$ rank	= D_INV

Table 33

Invariant Type	Function	Invariant To	Used For
Metric invariant ( $\Delta_{g}$ )	Construct metric	Geometric group action	$g$
Task invariant ( $\Delta_{S}$ )	Construct energy	Task-related nuisance	$S_{\text{task}}$

Table 34

Nuisance	Task Invariant
Brightness	Relative brightness, local contrast normalization
Contrast	Normalized gradient, structure tensor
Color	Lab chromaticity, chromaticity normalization
Texture	Low-frequency structure, multi-scale pyramid
Background	Center-surround contrast, saliency

Table 35

Condition	Paradigm	Applicable Scenario
Metric objectively determinable	Stage 0/1/2	Mass matrix known, Fisher estimable
Metric needs learning	Stage A/B/C	Nyström dissipation, sequence modeling G(z)
Metric uses unit/simplified engineering form	Unit metric + energy alignment	Case 2: Sim-to-Real, focus on energy target correctness

Table 36

Stage	What to Do	Training Object	Frozen	Uses flow	Uses energy
0	Train invariant head	$\Delta$	Detector/System	No	No
1	Statistical estimation $\Sigma$	None (statistical)	All	No	No
2	Train energy + decoder	energy, decoder	$\Sigma$ frozen	Yes	Yes

Table 37

Stage	Training Object	Loss	Purpose
A	encoder + metric + energy + decoder + $\eta$	Trajectory loss	Cold start
B	metric + energy + $\eta$	Residual loss	Fine-tune metric and energy
C	decoder	Trajectory loss	Align to task metric

Table 38

Loss	Jacobian	Contains state transition $\Phi(T,s)$	Amplified by Lyapunov
Trajectory	$J_{\rm traj}=\int_{0}^{T}\Phi(T,s)\partial_{\phi}f\,ds$	Yes	Yes
Strong-form residual	$J_{\rm strong,i}=-\partial_{\phi}f(z_{i})$	No	No
Weak-form residual	$J_{\rm weak,j}=\int\psi_{j}\partial_{\phi}f\,dt$	No	No

Table 39

Condition	Choice	Reason
Low observation noise	Strong form	Directly constrains residual, simple
High observation noise	Weak form	Avoids numerical differentiation error (Theorem 10.4)

Table 40

Scientific ML

Scientific machine learning combines data-driven models with scientific knowledge to analyze, predict, and simulate complex systems.

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SKG: A Law-Induced Engine from Inference to Induction

Jian An, Chaoshi Wu, Jinhui Wu

Physics-Informed ML Kernel Methods

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This paper proposes the SKG (Symplectic-Kernel Gradient) reasoning engine for law-induced inference and induction. Laws are a priori; data provide instances. Inference is geometric evolution under law constraints. We organize symplectic geometry, GENERIC, kernel methods, RKHS, group-equivariant networks, and constrained mechanics into a unified, analyzable, implementable, verifiable framework. We propose the SKG octuple S = (M, g, omega, H, S, G, Delta, Pi), whose four components correspond to four levels of laws: conservation laws, constitutive relations, symmetry groups, and axiom systems. Main results: (1) Invariants enter metric construction via the Jacobian: g = J^T Sigma^-1 J + epsilon_G I. For constant cases, we establish inverse reconstruction theory: solvability rank condition (Thm 3.9), consistency criterion (Thm 3.11), and uniqueness theorem (Thm 3.14). (2) The kernel bandwidth matrix is uniquely determined by Fisher information and satisfies task-independence (Thm 3.19). (3) The metric reconstruction error decomposes into four terms (Thm 3.21), with squared condition number kappa(Sigma^*)^2 as the dominant amplification factor (Thm 3.22). (4) For state-dependent cases, we establish function-space reconstruction theory on matrix-valued RKHS (Thms 4.8, 4.9). (5) Three-layer projections handle axiom, symplectic-step, and conservation constraints. (6) The residual equation yields the observable formula, four-dimensional rank condition, and noise propagation, while g_qq is structurally unidentifiable. (7) Kernel parameterization yields the RKHS extrapolation bound (Thm 4.15) and strict OOD gain theorem in group directions (Thm 8.15). (8) The learning-decoupling theorem (Thm 9.1) proves the sole object of induction is the parameters of the task energy. Validation: 1D damped oscillator and 2D rotation-invariant system; 6 unit tests, 10 experiments (paired t-test, p < 0.01).

Abstract

Scientific ML

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