Physically-Grounded Manifold Projection (PGMP)

• PGMP is a framework that projects data onto learned manifolds encapsulating physical, geometric, and semantic constraints to enhance model fidelity.
• Its methodology involves manifold learning, iterative optimization, and integration with existing models across robotics, biomolecular modeling, imaging, and spatial analytics.
• Empirical results demonstrate that PGMP achieves tighter theoretical error bounds and significant performance improvements compared to unconstrained approaches.

Physically-Grounded Manifold Projection (PGMP) is a unified framework for embedding physical structure and constraints into data-driven models via projection onto manifolds learned from empirical data and/or physical principles. PGMP encompasses methods that regularize model outputs by enforcing, during training or inference, adherence to submanifolds that encode the geometric, physical, or semantic rules inherent to the real-world task. This technique is applied across robotics (manipulation under mechanical constraints), biomolecular modeling (structure feasibility), medical imaging (anatomic plausibility restoration), cross-modal alignment (vision-language), and spatial analytics (geographic manifolds), yielding tighter theoretical error bounds and improved empirical performance.

1. Formal Definition and Core Principles

A physically-grounded manifold $\mathcal{M}$ is a subspace of the ambient state or data space, derived from constraints imposed by physics, mechanics, or semantics and estimated from empirical observations or simulation. For a model output $x \in \mathbb{R}^N$, PGMP seeks a projection $x_\mathrm{proj}$ onto $\mathcal{M}$ such that $x_\mathrm{proj}$ preserves fidelity to the original data while strictly respecting physical/semantic rules.

• In robotic manipulation, the manifold $\mathcal{T}\subset SE(3)$ represents feasible grasped-object poses defined by expert demonstrations, typically characterized by reduced DOFs (e.g., pure screw motion in nut threading) (Bogert et al., 3 Dec 2025).
• In molecular modeling, $\mathcal{M}$ encompasses all-atom structures devoid of steric clashes, geometric, and chemical infeasibilities enforced by explicit constraints (bond lengths, angles, chirality) (Chen et al., 10 Oct 2025).
• For medical artifact reduction, $\mathcal{M}$ encompasses artifact-free anatomical images—learned from simulations or patient ground truth—and may be coupled with foundation model priors for semantic plausibility (Li et al., 30 Dec 2025).
• In cross-modal alignment, $\mathcal{M}$ is a latent space where embeddings from different modalities (e.g., RGB-D and language) are aligned using shared physical ground-truth (object identity, referent) (Nguyen et al., 2020).
• In spatial networks, $\mathcal{M}$ represents a low-dimensional Euclidean manifold embedding locations such that friction-normalized distances preserve connectivity and diffusion phenomena (Jiang et al., 2024).

2. Mathematical Framework

PGMP formalizes projection onto physically-grounded manifolds through optimization principles tailored to each domain.

General PGMP Objective

For observed, corrupted, or provisional data $y$, one seeks:

$$\hat{x} = \arg\min_{x \in \mathcal{M}} \; \|\mathcal{F}(x) - y\|_2^2 + R(x)$$

where $\mathcal{F}$ denotes a forward model (e.g., imaging physics, kinematics), $R(x)$ encodes regularization (semantic, structural, or physical), and manifold membership enforces hard or soft constraints.

Robotics (GOMP)

The output pose $T_{so}^{\mathrm{pred}}$ is projected onto a manifold $\mathcal{T}$ derived via PCA/principal geodesic analysis of expert trajectories:

• Tangent coordinates: $\xi^*_t = \log(\bar{T}^{-1}T_{so}^t)$
• Manifold projection: Zeroing components orthogonal to the $d$-dim principal directions, exponential map back to $SE(3)$
• Objective: Behavioral cloning loss augmented with a manifold adherence penalty (Bogert et al., 3 Dec 2025).

Biomolecular Modeling

Constrained optimization of all-atom coordinates $\widetilde{x}$ onto the manifold of physically valid structures: $$x^* = \arg\min_x \tfrac{1}{2}\|x-\widetilde{x}\|^2 + \tfrac{1}{2}C(x)^T \alpha^{-1} C(x)$$ where $C_j(x)=0$ are the local, differentiable constraints; solved via Gauss-Seidel updates (Chen et al., 10 Oct 2025).

Medical Imaging

Manifold projection via direct $x$-prediction (DMP-Former): learn $f_\theta(y,C)\rightarrow \mathcal{M}$, such that output approximates clean anatomy subject to physics-based and semantic priors; optimization over

$$L_\text{total} = L_\text{manifold} + \lambda_\text{edge} L_\text{edge} + \lambda_\text{SSA} L_\text{SSA}$$

(Li et al., 30 Dec 2025).

Cross-modal Alignment

Triplet loss aligns image and text features in a shared latent manifold: $$L_{\mathrm{triplet}}(x_a, x_p, x_n) = \max\left\{ d(f_{\mu(a)}(x_a), f_{\mu(p)}(x_p)) - d(f_{\mu(a)}(x_a), f_{\mu(n)}(x_n)) + \alpha,\; 0 \right\}$$ with an optional Procrustes alignment for rigid fine-tuning (Nguyen et al., 2020).

Geographic Manifold

Transformation of raw interaction data through an inverse friction metric, followed by Isomap/t-SNE MDS embedding to recover a Euclidean manifold (Jiang et al., 2024).

3. Algorithmic Implementations

PGMP is instantiated via domain-specific procedures and often includes the following steps:

1. Manifold Learning: Extract empirical data (demonstrations, simulation, actual observations); compute low-dimensional representations (PCA, principal geodesics, embedding MLPs, Isomap/t-SNE).
2. Projection: Apply optimization (analytical or iterative, e.g., cyclic Gauss-Seidel for constraints, nearest-point geodesics for pose, direct neural mapping for images/text).
3. Adaptive Dimension Selection: In applications with ambiguous manifold dimensionality (e.g., assembly tasks), n-arm bandit mechanisms select the optimal projection dimension online based on observed success (Bogert et al., 3 Dec 2025).
4. Integration with Primary Model: PGMP generally augments (not replaces) existing learned policies or generative models via a lightweight module (projection head, alignment loss, etc.).
5. End-to-end Differentiability: For neural models, gradient computation through projection may use implicit differentiation approaches (e.g., conjugate gradients for biomolecular constraints) (Chen et al., 10 Oct 2025).

4. Theoretical Analysis: Error Bounds and Guarantees

PGMP provides strict theoretical improvements in error accumulation and physical validity relative to models unconstrained by physical manifolds.

• Imitation Learning (Robotics): Classical cost-to-go bound $J(\pi) \leq J(\pi^*) + nT^2\epsilon$ (Ross et al. 2011) is reduced under PGMP to $J(\pi_{\text{PGMP}}) \leq J(\pi^*) + dT^2\epsilon$, with $d \ll n$ (manifold dimension) (Bogert et al., 3 Dec 2025).
• Diffusion/Generative Models: Projection ensures every output conforms to hard physical constraints, producing 100% physically valid structures, as opposed to 20–60% invalidity in unconstrained baselines (Chen et al., 10 Oct 2025).
• Spatial Embedding: Existence of low-dimensional, homogeneous Euclidean embeddings for real geographic networks is demonstrated via combinatorial simplicial-complex analysis and rank-size/statistical physics arguments; almost all local neighborhoods are of ≤2 dimensions (Jiang et al., 2024).
• Error Correction: Projection actively mitigates error propagation along unconstrained DOFs or directions, preventing catastrophic drift in robotics and structure generation.

5. Experimental Results and Quantitative Metrics

PGMP has yielded robust empirical improvements in multiple domains.

Robotic Manipulation (Bogert et al., 3 Dec 2025)

Task	Demos	DP Baseline	GOMP
Nut-Threading	10	0.25	0.62
	100	0.26	0.78
Peg-Insertion	10	0.10	0.35
USB-Insertion	10	0.40	0.55
Battery-Cover	10	0.05	0.42

Success rates double or triple versus baselines, with qualitative improvements in object orientation and recovery from disturbances.

Biomolecular Modeling (Chen et al., 10 Oct 2025)

• PGMP (2-step) achieves complex-LDDT ≈ 0.62–0.94, matching 200-step baselines.
• Physical validity 100% versus up to 60% invalid for non-projected models.
• Inference speedup of ~10× (median) with full physical correctness.

Medical Imaging (Li et al., 30 Dec 2025)

• PGMP overall: PSNR = 36.80 dB, SSIM = 0.9165, Dice = 0.929±0.086; outperforming diffusion models by large margins.
• Inference: deterministic, 25 ms/slice versus ≥1.25 s/slice for diffusion.
• Segmentation Dice improvement up to +5.4% in unseen patients.

Cross-modal Alignment (Nguyen et al., 2020)

• Grounded-language micro-F1 = 0.983, macro-F1 = 0.725.
• Manifold metrics: MRR = 0.802, 5-NN accuracy = 0.787, distance correlation = 0.686.
• Outperforms CCA and deep CCA baselines.

Geographic Analytics (Jiang et al., 2024)

• PGMP manifold yields improvements in location uniformity (nearest-neighbor ratio rises from 1.04 to 2.27).
• Propagation modeling $R^2$ improves from 0.54 to 0.69.
• Isotropy and regularity confirmed by regression and grouping tests.

6. Generalization, Extensions, and Domain-Specific Variants

PGMP's architectural flexibility enables adaptation to diverse tasks:

• Beyond Rigid Pose Spaces: PGMP generalizes to manifolds over spaces combining positions, forces, contact wrenches, and compliant interaction parameters (Bogert et al., 3 Dec 2025).
• Spatio-Temporal Manifolds: Dynamic tasks may benefit from learning manifolds over sequences (dynamic mode decomposition, sequential autoencoders).
• Partial Observability: Robust manifold embedding from incomplete or noisy sensor data can be attained via learned observation-to-manifold mapping.
• Continuous Adaptation: Manifolds may drift in nonstationary environments; online or meta-learning adapts $\mathcal{M}$ in response to new dynamics or distributions.
• Semantic-Physical Coupling: Integration with large foundation models or medical prior networks provides semantic plausibility alongside physical constraints, exemplified by the SSA module in dental CBCT restoration (Li et al., 30 Dec 2025).
• Cross-modal/Multimodal Alignment: PGMP harmonizes data from multiple sensory sources (vision, language, depth, etc.), facilitating robust retrieval and classification in grounded language and perception (Nguyen et al., 2020).
• Spatial Analytics: In geography, PGMP is realized as distance normalization and manifold embedding techniques that regularize spatial interaction analysis, optimizing facility location and diffusion models (Jiang et al., 2024).

7. Limitations and Open Challenges

• Dependence on Data Quality: PGMP performance is sensitive to quality and representativity of expert demonstrations, simulation pairs, or extracted features in each domain.
• Negative Sampling and Margin Sensitivity: In cross-modal alignment, supervised triplet sampling and choice of triplet margin $\alpha$ influence alignment fidelity (Nguyen et al., 2020).
• Ambiguity in Semantic Descriptions: Accuracy may degrade with ambiguous or underspecified textual referents.
• Manifold Drift and Temporal Adaptivity: Real-world environments may induce manifold drift requiring continual re-learning.
• Scalability to Complex Constraint Spaces: Extension to high-dimensional or hybrid (pose × force) manifolds presents computational and modeling challenges.

A plausible implication is that further development of online, adaptive manifold learning and robust cross-modal representations will extend PGMP to a broader range of complex, real-world tasks with evolving constraints.