Christine Allen-Blanchette

Justice Mason, Christine Allen-Blanchette, Nicholas Zolman et al.

In many real-world settings, image observations of freely rotating 3D rigid bodies, such as satellites, may be available when low-dimensional measurements are not. However, the high-dimensionality of image data precludes the use of classical estimation techniques to learn the dynamics and a lack of interpretability reduces the usefulness of standard deep learning methods. In this work, we present a physics-informed neural network model to estimate and predict 3D rotational dynamics from image sequences. We achieve this using a multi-stage prediction pipeline that maps individual images to a latent representation homeomorphic to $\mathbf{SO}(3)$, computes angular velocities from latent pairs, and predicts future latent states using the Hamiltonian equations of motion with a learned representation of the Hamiltonian. We demonstrate the efficacy of our approach on a new rotating rigid-body dataset with sequences of rotating cubes and rectangular prisms with uniform and non-uniform density.

Dongzhe Zheng, Tao Zhong, Christine Allen-Blanchette

In this paper, we study solution operators of physical field equations on geometric meshes from a function-space perspective. We reveal that Hodge orthogonality fundamentally resolves spectral interference by isolating unlearnable topological degrees of freedom from learnable geometric dynamics, enabling an additive approximation confined to structure-preserving subspaces. Building on Hodge theory and operator splitting, we derive a principled operator-level decomposition. The result is a Hybrid Eulerian-Lagrangian architecture with an algebraic-level inductive bias we call Hodge Spectral Duality (HSD). In our framework, we use discrete differential forms to capture topology-dominated components and an orthogonal auxiliary ambient space to represent complex local dynamics. Our method achieves superior accuracy and efficiency on geometric graphs with enhanced fidelity to physical invariants. Our code is available at https://github.com/ContinuumCoder/Hodge-Spectral-Duality

Tao Zhong, Yixun Hu, Dongzhe Zheng et al. · princeton

We propose Neural Field Thermal Tomography (NeFTY), a differentiable physics framework for the quantitative 3D reconstruction of material properties from transient surface temperature measurements. While traditional thermography relies on pixel-wise 1D approximations that neglect lateral diffusion, and soft-constrained Physics-Informed Neural Networks (PINNs) often fail in transient diffusion scenarios due to gradient stiffness, NeFTY parameterizes the 3D diffusivity field as a continuous neural field optimized through a rigorous numerical solver. By leveraging a differentiable physics solver, our approach enforces thermodynamic laws as hard constraints while maintaining the memory efficiency required for high-resolution 3D tomography. Our discretize-then-optimize paradigm effectively mitigates the spectral bias and ill-posedness inherent in inverse heat conduction, enabling the recovery of subsurface defects at arbitrary scales. Experimental validation on synthetic data demonstrates that NeFTY significantly improves the accuracy of subsurface defect localization over baselines. Additional details at https://cab-lab-princeton.github.io/nefty/

Yulong Yang, Felix O'Mahony, Christine Allen-Blanchette

In this paper, we introduce group convolutional neural networks (GCNNs) equivariant to color variation. GCNNs have been designed for a variety of geometric transformations from 2D and 3D rotation groups, to semi-groups such as scale. Despite the improved interpretability, accuracy and generalizability of these architectures, GCNNs have seen limited application in the context of perceptual quantities. Notably, the recent CEConv network uses a GCNN to achieve equivariance to hue transformations by convolving input images with a hue rotated RGB filter. However, this approach leads to invalid RGB values which break equivariance and degrade performance. We resolve these issues with a lifting layer that transforms the input image directly, thereby circumventing the issue of invalid RGB values and improving equivariance error by over three orders of magnitude. Moreover, we extend the notion of color equivariance to include equivariance to saturation and luminance shift. Our hue-, saturation-, luminance- and color-equivariant networks achieve strong generalization to out-of-distribution perceptual variations and improved sample efficiency over conventional architectures. We demonstrate the utility of our approach on synthetic and real world datasets where we consistently outperform competitive baselines.

Justice Mason, Christine Allen-Blanchette, Nicholas Zolman et al.

In many real-world settings, image observations of freely rotating 3D rigid bodies may be available when low-dimensional measurements are not. However, the high-dimensionality of image data precludes the use of classical estimation techniques to learn the dynamics. The usefulness of standard deep learning methods is also limited, because an image of a rigid body reveals nothing about the distribution of mass inside the body, which, together with initial angular velocity, is what determines how the body will rotate. We present a physics-based neural network model to estimate and predict 3D rotational dynamics from image sequences. We achieve this using a multi-stage prediction pipeline that maps individual images to a latent representation homeomorphic to $\mathbf{SO}(3)$, computes angular velocities from latent pairs, and predicts future latent states using the Hamiltonian equations of motion. We demonstrate the efficacy of our approach on new rotating rigid-body datasets of sequences of synthetic images of rotating objects, including cubes, prisms and satellites, with unknown uniform and non-uniform mass distributions. Our model outperforms competing baselines on our datasets, producing better qualitative predictions and reducing the error observed for the state-of-the-art Hamiltonian Generative Network by a factor of 2.

Christine Allen-Blanchette

A growing body of work leverages the Hamiltonian formalism as an inductive bias for physically plausible neural network based video generation. The structure of the Hamiltonian ensures conservation of a learned quantity (e.g., energy) and imposes a phase-space interpretation on the low-dimensional manifold underlying the input video. While this interpretation has the potential to facilitate the integration of learned representations in downstream tasks, existing methods are limited in their applicability as they require a structural prior for the configuration space at design time. In this work, we present a GAN-based video generation pipeline with a learned configuration space map and Hamiltonian neural network motion model, to learn a representation of the configuration space from data. We train our model with a physics-inspired cyclic-coordinate loss function which encourages a minimal representation of the configuration space and improves interpretability. We demonstrate the efficacy and advantages of our approach on the Hamiltonian Dynamics Suite Toy Physics dataset.

Fynn Fromme, Hans Harder, Christine Allen-Blanchette et al.

The use of machine learning for modeling, understanding, and controlling large-scale physics systems is quickly gaining in popularity, with examples ranging from electromagnetism over nuclear fusion reactors and magneto-hydrodynamics to fluid mechanics and climate modeling. These systems - governed by partial differential equations - present unique challenges regarding the large number of degrees of freedom and the complex dynamics over many scales both in space and time, and additional measures to improve accuracy and sample efficiency are highly desirable. We present an end-to-end equivariant surrogate model consisting of an equivariant convolutional autoencoder and an equivariant convolutional LSTM using $G$-steerable kernels. As a case study, we consider the three-dimensional Rayleigh-Bénard convection, which describes the buoyancy-driven fluid flow between a heated bottom and a cooled top plate. While the system is E(2)-equivariant in the horizontal plane, the boundary conditions break the translational equivariance in the vertical direction. Our architecture leverages vertically stacked layers of $D_4$-steerable kernels, with additional partial kernel sharing in the vertical direction for further efficiency improvement. We demonstrate significant gains in sample and parameter efficiency, as well as a better scaling to more complex dynamics. The accompanying code is available under https://github.com/FynnFromme/equivariant-rb-forecasting.

Zhixuan Zhao, Tao Zhong, Yixun Hu et al.

Inverse problems in scientific sensing are often solved with either hand-designed regularizers or supervised networks trained on simulated labels, yet both can fail when the forward model is nonlinear, spectrally coupled, and physically delicate. We study this issue for noise sensing based on nitrogen-vacancy (NV) centers in diamond, where a quantum sensor measures magnetic-noise spectra generated by sparse spin sources. We show that replacing a common scalar/coherent forward approximation with a tensor power-summed dipolar operator changes the inverse landscape and exposes a center-collapse failure mode in free-density optimization. We propose NeTMY, an amortization-free coordinate neural field coupled to the differentiable NV forward model, with annealed positional encoding, multiscale optimization, sparsity/gating, and spectrum-fidelity losses. Across sparse synthetic reconstructions generated by the corrected operator, NeTMY achieves the best localization and distributional metrics in the tested benchmark. Mechanism experiments show that NeTMY does not directly execute the raw density-space gradient; its parameterization smooths and redistributes updates, mitigating the center-collapse pathology. These results position NV quantum sensing as a useful testbed for physics-faithful neural inverse problems.

Tao Zhong, Dongzhe Zheng, Christine Allen-Blanchette

Sparse Mixture-of-Experts (MoE) layers route tokens through a handful of experts, and learning-free compression of these layers reduces inference cost without retraining. A subtle obstruction blocks every existing compressor in this family: three experts can each be pairwise compatible yet form an irreducible cycle when merged together, so any score that ranks experts on pairwise signals is structurally blind to which triples are jointly mergeable. We show the obstruction is a precise mathematical object, the harmonic kernel of the simplicial Laplacian on a 2-complex whose vertices are experts, whose edges carry KL merge barriers, and whose faces carry triplet barriers; Hodge-decomposing the edge-barrier signal isolates the kernel exactly. We turn the diagnostic into a selection objective: HodgeCover greedily covers the harmonic-critical edges and triplet-critical triangles, and a hybrid variant of HodgeCover pairs it with off-the-shelf weight pruning on survivors. On three open-weight Sparse MoE backbones under aggressive expert reduction, HodgeCover matches state-of-the-art learning-free baselines on the expert-reduction axis, leads on the aggressive-compression frontier of the hybrid axis, and uniquely balances retained mass across all four Hodge components. These results show that exposing the harmonic kernel of a learned MoE structure changes which compressor wins at the regime that matters most.

Yulong Yang, Zhikun Xu, Yaojun Li et al.

When the color distribution of input images changes at inference, the performance of conventional neural network architectures drops considerably. A few researchers have begun to incorporate prior knowledge of color geometry in neural network design. These color equivariant architectures have modeled hue variation with 2D rotations, and saturation and luminance transformations as 1D translations. While this approach improves neural network robustness to color variations in a number of contexts, we find that approximating saturation and luminance (interval valued quantities) as 1D translations introduces appreciable artifacts. In this paper, we introduce a color equivariant architecture that is truly equivariant. Instead of approximating the interval with the real line, we lift values on the interval to values on the circle (a double-cover) and build equivariant representations there. Our approach resolves the approximation artifacts of previous methods, improves interpretability and generalizability, and achieves better predictive performance than conventional and equivariant baselines on tasks such as fine-grained classification and medical imaging tasks. Going beyond the context of color, we show that our proposed lifting can also extend to geometric transformations such as scale.

Tao Zhong, Christine Allen-Blanchette · princeton

We propose GAGrasp, a novel framework for dexterous grasp generation that leverages geometric algebra representations to enforce equivariance to SE(3) transformations. By encoding the SE(3) symmetry constraint directly into the architecture, our method improves data and parameter efficiency while enabling robust grasp generation across diverse object poses. Additionally, we incorporate a differentiable physics-informed refinement layer, which ensures that generated grasps are physically plausible and stable. Extensive experiments demonstrate the model's superior performance in generalization, stability, and adaptability compared to existing methods. Additional details at https://gagrasp.github.io/

Yaojun Li, Yulong Yang, Christine Allen-Blanchette

While Hamiltonian mechanics provides a powerful inductive bias for neural networks modeling dynamical systems, Hamiltonian Neural Networks and their variants often fail to capture complex temporal dynamics spanning multiple timescales. This limitation is commonly linked to the spectral bias of deep neural networks, which favors learning low-frequency, slow-varying dynamics. Prior approaches have sought to address this issue through symplectic integration schemes that enforce energy conservation or by incorporating geometric constraints to impose structure on the configuration-space. However, such methods either remain limited in their ability to fully capture multiscale dynamics or require substantial domain specific assumptions. In this work, we exploit the observation that Hamiltonian functions admit decompositions into explicit fast and slow modes and can be reconstructed from these components. We introduce the Frequency-Separable Hamiltonian Neural Network (FS-HNN), which parameterizes the system Hamiltonian using multiple networks, each governed by Hamiltonian dynamics and trained on data sampled at distinct timescales. We further extend this framework to partial differential equations by learning a state- and boundary-conditioned symplectic operators. Empirically, we show that FS-HNN improves long-horizon extrapolation performance on challenging dynamical systems and generalizes across a broad range of ODE and PDE problems.

Tao Zhong, Jonah Buchanan, Christine Allen-Blanchette · princeton

We propose a new approach to vision-based dexterous grasp translation, which aims to transfer grasp intent across robotic hands with differing morphologies. Given a visual observation of a source hand grasping an object, our goal is to synthesize a functionally equivalent grasp for a target hand without requiring paired demonstrations or hand-specific simulations. We frame this problem as a stochastic transport between grasp distributions using the Schrödinger Bridge formalism. Our method learns to map between source and target latent grasp spaces via score and flow matching, conditioned on visual observations. To guide this translation, we introduce physics-informed cost functions that encode alignment in base pose, contact maps, wrench space, and manipulability. Experiments across diverse hand-object pairs demonstrate our approach generates stable, physically grounded grasps with strong generalization. This work enables semantic grasp transfer for heterogeneous manipulators and bridges vision-based grasping with probabilistic generative modeling. Additional details at https://grasp2grasp.github.io/

Jiayin Liu, Yulong Yang, Vineet Bansal et al.

From metronomes to celestial bodies, mechanics underpins how the world evolves in time and space. With consideration of this, a number of recent neural network models leverage inductive biases from classical mechanics to encourage model interpretability and ensure forecasted states are physical. However, in general, these models are designed to capture the dynamics of a single system with fixed physical parameters, from state-space measurements of a known configuration space. In this paper we introduce Symplectic Phase Space GAN (SPS-GAN) which can capture the dynamics of multiple systems, and generalize to unseen physical parameters from. Moreover, SPS-GAN does not require prior knowledge of the system configuration space. In fact, SPS-GAN can discover the configuration space structure of the system from arbitrary measurement types (e.g., state-space measurements, video frames). To achieve physically plausible generation, we introduce a novel architecture which embeds a Hamiltonian neural network recurrent module in a conditional GAN backbone. To discover the structure of the configuration space, we optimize the conditional time-series GAN objective with an additional physically motivated term to encourages a sparse representation of the configuration space. We demonstrate the utility of SPS-GAN for trajectory prediction, video generation and symmetry discovery. Our approach captures multiple systems and achieves performance on par with supervised models designed for single systems.

Keqin Wang, Yulong Yang, Ishan Saha et al.

While graph neural networks (GNNs) have allowed researchers to successfully apply neural networks to non-Euclidean domains, deep GNNs often exhibit lower predictive performance than their shallow counterparts. This phenomena has been attributed in part to oversmoothing, the tendency of node representations to become increasingly similar with network depth. In this paper we introduce an analogy between oversmoothing in GNNs and consensus (i.e., perfect agreement) in the opinion dynamics literature. We show that the message passing algorithms of several GNN models are equivalent to linear opinion dynamics models which have been shown to converge to consensus for all inputs regardless of the graph structure. This new perspective on oversmoothing motivates the use of nonlinear opinion dynamics as an inductive bias in GNN models. In our Behavior-Inspired Message Passing (BIMP) GNN, we leverage the nonlinear opinion dynamics model which is more general than the linear opinion dynamics model, and can be designed to converge to dissensus for general inputs. Through extensive experiments we show that BIMP resists oversmoothing beyond 100 time steps and consistently outperforms existing architectures even when those architectures are amended with oversmoothing mitigation techniques. We also show that BIMP has several desirable properties including well behaved gradients and adaptability to homophilic and heterophilic datasets.

Yulong Yang, Bowen Feng, Keqin Wang et al.

From pedestrians to Kuramoto oscillators, interactions between agents govern how dynamical systems evolve in space and time. Discovering how these agents relate to each other has the potential to improve our understanding of the often complex dynamics that underlie these systems. Recent works learn to categorize relationships between agents based on observations of their physical behavior. These approaches model relationship categories as outcomes of a categorical distribution which is limiting and contrary to real-world systems, where relationship categories often intermingle and interact. In this work, we introduce a level of abstraction between the observable behavior of agents and the latent categories that determine their behavior. To do this, we learn a mapping from agent observations to agent preferences for a set of latent categories. The learned preferences and inter-agent proximity are integrated in a nonlinear opinion dynamics model, which allows us to naturally identify mutually exclusive categories, predict an agent's evolution in time, and control an agent's behavior. Through extensive experiments, we demonstrate the utility of our model for learning interpretable categories, and the efficacy of our model for long-horizon trajectory prediction.

Yulong Yang, Felix O'Mahony, Christine Allen-Blanchette

In this paper, we introduce group convolutional neural networks (GCNNs) equivariant to color variation. GCNNs have been designed for a variety of geometric transformations from 2D and 3D rotation groups, to semi-groups such as scale. Despite the improved interpretability, accuracy and generalizability of these architectures, GCNNs have seen limited application in the context of perceptual quantities. Notably, the recent CEConv network uses a GCNN to achieve equivariance to hue transformations by convolving input images with a hue rotated RGB filter. However, this approach leads to invalid RGB values which break equivariance and degrade performance. We resolve these issues with a lifting layer that transforms the input image directly, thereby circumventing the issue of invalid RGB values and improving equivariance error by over three orders of magnitude. Moreover, we extend the notion of color equivariance to include equivariance to saturation and luminance shift. Our hue-, saturation-, luminance- and color-equivariant networks achieve strong generalization to out-of-distribution perceptual variations and improved sample efficiency over conventional architectures. We demonstrate the utility of our approach on synthetic and real world datasets where we consistently outperform competitive baselines.

Christine Allen-Blanchette, Kostas Daniilidis

Images encode both the state of the world and its content. The former is useful for tasks such as planning and control, and the latter for classification. The automatic extraction of this information is challenging because of the high-dimensionality and entangled encoding inherent to the image representation. This article introduces two theoretical approaches aimed at the resolution of these challenges. The approaches allow for the interpolation and extrapolation of images from an image sequence by joint estimation of the image representation and the generators of the sequence dynamics. In the first approach, the image representations are learned using probabilistic PCA \cite{tipping1999probabilistic}. The linear-Gaussian conditional distributions allow for a closed form analytical description of the latent distributions but assumes the underlying image manifold is a linear subspace. In the second approach, the image representations are learned using probabilistic nonlinear PCA which relieves the linear manifold assumption at the cost of requiring a variational approximation of the latent distributions. In both approaches, the underlying dynamics of the image sequence are modelled explicitly to disentangle them from the image representations. The dynamics themselves are modelled with Lie group structure which enforces the desirable properties of smoothness and composability of inter-image transformations.

Christine Allen-Blanchette, Sushant Veer, Anirudha Majumdar et al.

The dominant paradigms for video prediction rely on opaque transition models where neither the equations of motion nor the underlying physical quantities of the system are easily inferred. The equations of motion, as defined by Newton's second law, describe the time evolution of a physical system state and can therefore be applied toward the determination of future system states. In this paper, we introduce a video prediction model where the equations of motion are explicitly constructed from learned representations of the underlying physical quantities. To achieve this, we simultaneously learn a low-dimensional state representation and system Lagrangian. The kinetic and potential energy terms of the Lagrangian are distinctly modelled and the low-dimensional equations of motion are explicitly constructed using the Euler-Lagrange equations. We demonstrate the efficacy of this approach for video prediction on image sequences rendered in modified OpenAI gym Pendulum-v0 and Acrobot environments.

Carlos Esteves, Yinshuang Xu, Christine Allen-Blanchette et al.

Several popular approaches to 3D vision tasks process multiple views of the input independently with deep neural networks pre-trained on natural images, achieving view permutation invariance through a single round of pooling over all views. We argue that this operation discards important information and leads to subpar global descriptors. In this paper, we propose a group convolutional approach to multiple view aggregation where convolutions are performed over a discrete subgroup of the rotation group, enabling, thus, joint reasoning over all views in an equivariant (instead of invariant) fashion, up to the very last layer. We further develop this idea to operate on smaller discrete homogeneous spaces of the rotation group, where a polar view representation is used to maintain equivariance with only a fraction of the number of input views. We set the new state of the art in several large scale 3D shape retrieval tasks, and show additional applications to panoramic scene classification.

Carlos Esteves, Christine Allen-Blanchette, Ameesh Makadia et al.

We address the problem of 3D rotation equivariance in convolutional neural networks. 3D rotations have been a challenging nuisance in 3D classification tasks requiring higher capacity and extended data augmentation in order to tackle it. We model 3D data with multi-valued spherical functions and we propose a novel spherical convolutional network that implements exact convolutions on the sphere by realizing them in the spherical harmonic domain. Resulting filters have local symmetry and are localized by enforcing smooth spectra. We apply a novel pooling on the spectral domain and our operations are independent of the underlying spherical resolution throughout the network. We show that networks with much lower capacity and without requiring data augmentation can exhibit performance comparable to the state of the art in standard retrieval and classification benchmarks.

Carlos Esteves, Christine Allen-Blanchette, Xiaowei Zhou et al.

Convolutional neural networks (CNNs) are inherently equivariant to translation. Efforts to embed other forms of equivariance have concentrated solely on rotation. We expand the notion of equivariance in CNNs through the Polar Transformer Network (PTN). PTN combines ideas from the Spatial Transformer Network (STN) and canonical coordinate representations. The result is a network invariant to translation and equivariant to both rotation and scale. PTN is trained end-to-end and composed of three distinct stages: a polar origin predictor, the newly introduced polar transformer module and a classifier. PTN achieves state-of-the-art on rotated MNIST and the newly introduced SIM2MNIST dataset, an MNIST variation obtained by adding clutter and perturbing digits with translation, rotation and scaling. The ideas of PTN are extensible to 3D which we demonstrate through the Cylindrical Transformer Network.