Deep-learning models of anatomy can be numerically plausible yet anatomically impossible, and they generalize poorly when data are scarce. We introduce Anatomy-Informed Neural Networks (AINN), in which soft anatomic priors enter as penalty terms in the loss (e.g., a branching penalty that treats a renal transplant artery off the iliac instead of the aorta as unexpected rather than impossible), in direct analogy to a physics-informed neural network, and hard anatomic priors (e.g., continuity of the vessel) are built into the architecture and state representation, making such invalid predictions impossible by construction wherever the prior admits architectural enforcement. We develop it on a clinical test case with limited data: how the aortoiliac tree deforms when a stiff wire is introduced endoluminally. This is important to contemporary aortic surgery and will matter to autonomous endovascular navigation. We lift the vessel centerline and the wire path from R^3 to curves of frames in the Lie group SE(3), and couple a Cosserat-rod wire to a tortuosity-modulated, anatomically anchored vessel through a unilateral lumen-contact inequality. The prediction is a constrained minimizer of the coupled elastic energy, with contact forces as its Lagrange multipliers. Supervision is a Wasserstein-2 optimal-transport loss between the predicted projection through the C-arm geometry and the observed angiogram, so a 2D angiogram can train a 3D prediction. The kinematics, loss and projection are verified against known ground truth; the mechanics solver only against its own optimality conditions, and predicted displacement is not yet mesh-converged. Here, no network is trained. Future work will transfer this in silico model to real CT scans and test whether it improves predictive accuracy and reduces the training data required.
Generative modeling of protein backbones promises the de novo design of proteins with prescribed structural and functional properties. Existing diffusion and flow-matching models produce high-quality backbones on SE(3)^N, but inference requires numerically integrating an ODE over hundreds of network evaluations, each involving a Lie group exponential map - a bottleneck for high-throughput design campaigns. We introduce SE(3)-MeanFlow, a few-step generative framework that extends MeanFlow from Euclidean space to the Lie group geometry of protein frames. Working natively in the Lie algebra so(3) and in R^3, we derive closed-form average-velocity identities for rotations and translations, giving simulation-free training targets. We further introduce an SE(3) alpha-Flow objective that removes the Jacobian-vector product from the rotation branch and serves as a warm-up stage, after which training switches to a small-t stabilized MeanFlow loss that is used for the remainder of pretraining and for rectification-based post-training. In protein backbone generation, SE(3)-MeanFlow matches or exceeds flow-matching baselines that use several times more sampling steps, and its advantage widens in the few-step regime, where rectification lets it lead at every matched budget - at a modest cost in diversity.
3D Gaussian Splatting (3DGS) captures scenes by coupling explicit geometry (position, covariance) with view-dependent photometry (Spherical Harmonics). However, building $\mathrm{SE}(3)$-equivariant architectures on these primitives presents a fundamental representation bottleneck. Color has been treated as a signal rather than a geometric entity, making it nontrivial to unify symmetry across geometry and appearance as the camera frame changes. While translations are handled by relative coordinates, rotations act heterogeneously across attributes: $μ\mapsto Rμ$, $Σ\mapsto RΣR^\top$, and $f_\ell\mapsto D^\ell(R)f_\ell$. This mismatch complicates strict equivariance, leading existing methods to either discard or flatten SH coefficients, thereby breaking symmetry. We propose a unified solution rooted in representation theory: for SH degrees $\ell\le2$, photometry is algebraically isomorphic to a rank-2 geometric tensor. We prove that the Wigner-$D$ action on these SH coefficients can be exactly reformulated as the conjugation action on $3\times3$ matrices. Leveraging this, we introduce the Unified Matrix Embedding, a lifting that maps all Gaussian attributes into a unified carrier space, $\mathfrak{gl}(3)$. Building on the "Color-as-Geometry" formulation, we present E3DGS, a rigid-body ($\mathrm{SE}(3)$) equivariant architecture that processes 3D Gaussians without Clebsch-Gordan tensor products. Evaluations on object vision and action-conditioned Gaussian world modeling demonstrate that our unified approach yields strong robustness under camera-frame changes and improved data efficiency.