Abstract Cyclic animation is widely used in computer graphics and interactive content. It supports seamless playback in games, VR, and interactive simulation, where short clips must repeat smoothly over long durations. Converting an observed non‐cyclic trajectory into a seamless loop is challenging because the endpoint states of the observed sequence rarely match exactly, and the governing equations of the underlying system are often unavailable. We therefore propose an equation‐free framework that identifies a Koopman surrogate from the observed trajectory and computes a cyclic trajectory by applying a Fourier‐parameterized, time‐varying control input under a hard temporal periodicity constraint. The resulting formulation reduces cyclic synthesis to a linearly constrained quadratic program that can be solved efficiently through a structured KKT system. Our method is applicable to a diverse range of examples, including N‐body systems, cloth, deformable objects, shallow water, etc.

We present a low-rank Koopman operator formulation for accelerating deformable subspace simulation. Using a Dynamic Mode Decomposition (DMD) parameterization of the Koopman operator, our method learns the temporal evolution of deformable dynamics and predicts future states through efficient matrix evaluations instead of sequential time integration. This yields log-linear scaling in the number of time steps and allows large portions of the trajectory to be skipped while retaining accuracy. The resulting temporal efficiency is especially advantageous for optimization tasks such as control and initial-state estimation, where the objective often depends largely on the final configuration. To broaden the scope of Koopman-based reduced-order models in graphics, we introduce a discretization-agnostic extension that learns shared dynamic behavior across multiple shapes and mesh resolutions. Prior DMD-based approaches have been restricted to a single shape and discretization, which limits their usefulness for tasks involving geometry variation. Our formulation generalizes across both shape and discretization, which enables fast shape optimization that was previously impractical for DMD models. This expanded capability highlights the potential of Koopman operator learning as a practical tool for efficient deformable simulation and design.

Simulating large-scale articulated assemblies poses a significant challenge due to the numerical stiffness and geometric complexity of jointed structures. Conventional rigid body solvers struggle with the high nonlinearity induced by rotation parameterization. This difficulty becomes more pronounced for multiple two-way-coupled bodies. This paper introduces a novel framework that leverages the linear kinematic mapping of Affine Body Dynamics (ABD). As ABD targets near-rigid objects, the constitutive variations of different materials become negligible, which justifies a co-rotational approach to isolate geometric nonlinearities of the system. This insight enables the use of constant system matrices that can be pre-factorized throughout the simulation, even with fully implicit integration schemes. To manage the high DOF counts of large-scale systems, we map primal body coordinates onto a compact dual space defined by minimal joint degrees of freedom. By solving the resulting KKT systems, our method ensures exact constraint enforcement and physically accurate motion propagation. We provide a suite of specialized solvers tailored for diverse joint topologies, including chains, trees, closed loops, and irregular networks. Experimental results show that our approach achieves interactive rates for systems with hundreds of thousands of bodies on a single CPU core, while maintaining excellent stability at large time steps.

Differentiable simulation has become a powerful tool for system identification. While prior work has focused on identifying robot properties using robot-specific data or object properties using object-specific data, our approach calibrates object properties by using information from the robot, without relying on data from the object itself. Specifically, we utilize robot joint encoder information, which is commonly available in standard robotic systems. Our key observation is that by analyzing the robot's reactions to manipulated objects, we can infer properties of those objects, such as inertia and softness. Leveraging this insight, we develop differentiable simulations of robot-object interactions to inversely identify the properties of the manipulated objects. Our approach relies solely on proprioception – the robot's internal sensing capabilities – and does not require external measurement tools or vision-based tracking systems. This general method is applicable to any articulated robot and requires only joint position information. We demonstrate the effectiveness of our method on a low-cost robotic platform, achieving accurate mass and elastic modulus estimations of manipulated objects with just a few seconds of computation on a laptop.

The development of novel autonomous underwater gliders has been hindered by limited shape diversity, primarily due to the reliance on traditional design tools that depend heavily on manual trial and error. Building an automated design framework is challenging due to the complexities of representing glider shapes and the high computational costs associated with modeling complex solid-fluid interactions. In this work, we introduce an AI-enhanced automated computational framework designed to overcome these limitations by enabling the creation of underwater robots with non-trivial hull shapes. Our approach involves an algorithm that cooptimizes both shape and control signals, utilizing a reducedorder geometry representation and a differentiable neural-network-based fluid surrogate model. This end-to-end design workflow facilitates rapid iteration and evaluation of hydrodynamic performance, leading to the discovery of optimal and complex hull shapes across various control settings. We validate our method through wind tunnel experiments and swimming pool gliding tests, demonstrating that our computationally designed gliders surpass manually designed counterparts in terms of energy efficiency. By addressing challenges in efficient shape representation and neural fluid surrogate models, our work paves the way for the development of highly efficient underwater gliders, with implications for long-range ocean exploration and environmental monitoring.

Discontinuities in spatial derivatives appear in a wide range of physical systems, from creased thin sheets to materials with sharp stiffness transitions. Accurately modeling these features is essential for simulation but remains challenging for traditional mesh-based methods, which require discontinuity-aligned remeshing—entangling geometry with simulation and hindering generalization across shape families.

Eigenanalysis of differential operators, such as the Laplace operator or elastic energy Hessian, is typically restricted to a single shape and its discretization, limiting reduced order modeling (ROM). We introduce the first eigenanalysis method for continuously parameterized shape families. Given a parametric shape, our method constructs spatial neural fields that represent eigen-functions across the entire shape space. It is agnostic to the specific shape representation, requiring only an inside/outside indicator function that depends on shape parameters. Eigenfunctions are computed by minimizing a variational principle over nested spaces with orthogonality constraints. Since eigenvalues may swap dominance at points of multiplicity, we jointly train multiple eigenfunctions while dynamically reordering them based on their eigenvalues at each step. Through causal gradient filtering , this reordering is reflected in backpropagation. Our method enables applications to operate over shape space, providing a single ROM that encapsulates vibration modes for all shapes, including previously unseen ones. Since our eigenanalysis is differentiable with respect to shape parameters, it facilitates eigenfunction-aware shape optimization. We evaluate our approach on shape optimization for sound synthesis and locomotion, as well as reduced-order modeling for elastodynamic simulation.

We propose a hybrid neural network (NN) and PDE approach for learning generalizable PDE dynamics from motion observations. Many NN approaches learn an end-to-end model that implicitly models both the governing PDE and constitutive models (or material models). Without explicit PDE knowledge, these approaches cannot guarantee physical correctness and have limited generalizability. We argue that the governing PDEs are often well-known and should be explicitly enforced rather than learned. Instead, constitutive models are particularly suitable for learning due to their data-fitting nature. To this end, we introduce a new framework termed "Neural Constitutive Laws" (NCLaw), which utilizes a network architecture that strictly guarantees standard constitutive priors, including rotation equivariance and undeformed state equilibrium. We embed this network inside a differentiable simulation and train the model by minimizing a loss function based on the difference between the simulation and the motion observation. We validate NCLaw on various large-deformation dynamical systems, ranging from solids to fluids. After training on a single motion trajectory, our method generalizes to new geometries, initial/boundary conditions, temporal ranges, and even multi-physics systems. On these extremely out-of-distribution generalization tasks, NCLaw is orders-of-magnitude more accurate than previous NN approaches. Real-world experiments demonstrate our method’s ability to learn constitutive laws from videos.

Implicit Neural Spatial Representation (INSR) has emerged as an effective representation of spatially-dependent vector fields. This work explores solving time-dependent PDEs with INSR. Classical PDE solvers introduce both temporal and spatial discretizations. Common spatial discretizations include meshes and meshless point clouds, where each degree-of-freedom corresponds to a location in space. While these explicit spatial correspondences are intuitive to model and understand, these representations are not necessarily optimal for accuracy, memory usage, or adaptivity. Keeping the classical temporal discretization unchanged (e.g., explicit/implicit Euler), we explore INSR as an alternative spatial discretization, where spatial information is implicitly stored in the neural network weights. The network weights then evolve over time via time integration. Our approach does not require any training data generated by existing solvers because our approach is the solver itself. We validate our approach on various PDEs with examples involving large elastic deformations, turbulent fluids, and multi-scale phenomena. While slower to compute than traditional representations, our approach exhibits higher accuracy and lower memory consumption. Whereas classical solvers can dynamically adapt their spatial representation only by resorting to complex remeshing algorithms, our INSR approach is intrinsically adaptive. By tapping into the rich literature of classic time integrators, e.g., operator-splitting schemes, our method enables challenging simulations in contact mechanics and turbulent flows where previous neural-physics approaches struggle. Videos and codes are available on the project page: http://www.cs.columbia.edu/cg/INSR-PDE/

Efficient numerical solvers for partial differential equations empower science and engineering. One commonly employed numerical solver is the preconditioned conjugate gradient (PCG) algorithm, whose performance is largely affected by the preconditioner quality. However, designing high-performing preconditioner with traditional numerical methods is highly non-trivial, often requiring problem-specific knowledge and meticulous matrix operations. We present a new method that leverages learning-based approach to obtain an approximate matrix factorization to the system matrix to be used as a preconditioner in the context of PCG solvers. Our high-level intuition comes from the shared property between preconditioners and network-based PDE solvers that excels at obtaining approximate solutions at a low computational cost. Such observation motivates us to represent preconditioners as graph neural networks (GNNs). In addition, we propose a new loss function that rewrites traditional preconditioner metrics to incorporate inductive bias from PDE data distributions, enabling effective training of high-performing preconditioners. We conduct extensive experiments to demonstrate the efficacy and generalizability of our proposed approach on solving various 2D and 3D linear second-order PDEs.

Linear reduced-order modeling (ROM) simplifies complex simulations by approximating the behavior of a system using a simplified kinematic representation. Typically, ROM is trained on input simulations created with a specific spatial discretization, and then serves to accelerate simulations with the same discretization. This discretization-dependence is restrictive.

We propose a hybrid neural network and physics framework for reduced-order modeling of elastoplasticity and fracture. State-of-the-art scientific computing models like the Material Point Method (MPM) faithfully simulate large-deformation elastoplasticity and fracture mechanics. However, their long runtime and large memory consumption render them unsuitable for applications constrained by computation time and memory usage, e.g., virtual reality. To overcome these barriers, we propose a reduced-order framework. Our key innovation is training a low-dimensional manifold for the Kirchhoff stress field via an implicit neural representation. This low-dimensional neural stress field (NSF) enables efficient evaluations of stress values and, correspondingly, internal forces at arbitrary spatial locations. In addition, we also train neural deformation and affine fields to build low-dimensional manifolds for the deformation and affine momentum fields. These neural stress, deformation, and affine fields share the same low-dimensional latent space, which uniquely embeds the high-dimensional simulation state. After training, we run new simulations by evolving in this single latent space, which drastically reduces the computation time and memory consumption. Our general continuum-mechanics-based reduced-order framework is applicable to any phenomena governed by the elastodynamics equation. To showcase the versatility of our framework, we simulate a wide range of material behaviors, including elastica, sand, metal, non-Newtonian fluids, fracture, contact, and collision. We demonstrate dimension reduction by up to 100,000 × and time savings by up to 10 ×.

We propose a technique to simulate granular materials that exploits the dual strengths of discrete and continuum treatments. Discrete element simulations provide unmatched levels of detail and generality, but prove excessively costly when applied to large scale systems. Continuum approaches are computationally tractable, but limited in applicability due to built-in modeling assumptions; e.g., models suitable for granular flows typically fail to capture clogging, bouncing and ballistic motion. In our hybrid approach, an oracle dynamically partitions the domain into continuum regions where safe, and discrete regions where necessary. The domains overlap along transition zones, where a Lagrangian dynamics mass-splitting coupling principle enforces agreement between the two simulation states. Enrichment and homogenization operations allow the partitions to evolve over time. This approach accurately and efficiently simulates scenarios that previously required an entirely discrete treatment.