Joseph Teran

Chenfanfu Jiang, Craig Schroeder, Joseph Teran

We present a new technique for transferring momentum and velocity between particles and grid with Particle-In-Cell (PIC) calculations which we call Affine-Particle-In-Cell (APIC). APIC represents particle velocities as locally affine, rather than locally constant as in traditional PIC. We show that this representation allows APIC to conserve linear and angular momentum across transfers while also dramatically reducing numerical diffusion usually associated with PIC. Notably, conservation is achieved with lumped mass, as opposed to the more commonly used Fluid Implicit Particle (FLIP) transfers which require a 'full' mass matrix for exact conservation. Furthermore, unlike FLIP, APIC retains a filtering property of the original PIC and thus does not accumulate velocity modes on particles as FLIP does. In particular, we demonstrate that APIC does not experience velocity instabilities that are characteristic of FLIP in a number of Material Point Method (MPM) hyperelasticity calculations. Lastly, we demonstrate that when combined with the midpoint rule for implicit update of grid momentum that linear and angular momentum are exactly conserved.

Ayano Kaneda, Osman Akar, Jingyu Chen et al.

We present a novel deep learning approach to approximate the solution of large, sparse, symmetric, positive-definite linear systems of equations. These systems arise from many problems in applied science, e.g., in numerical methods for partial differential equations. Algorithms for approximating the solution to these systems are often the bottleneck in problems that require their solution, particularly for modern applications that require many millions of unknowns. Indeed, numerical linear algebra techniques have been investigated for many decades to alleviate this computational burden. Recently, data-driven techniques have also shown promise for these problems. Motivated by the conjugate gradients algorithm that iteratively selects search directions for minimizing the matrix norm of the approximation error, we design an approach that utilizes a deep neural network to accelerate convergence via data-driven improvement of the search directions. Our method leverages a carefully chosen convolutional network to approximate the action of the inverse of the linear operator up to an arbitrary constant. We train the network using unsupervised learning with a loss function equal to the $L^2$ difference between an input and the system matrix times the network evaluation, where the unspecified constant in the approximate inverse is accounted for. We demonstrate the efficacy of our approach on spatially discretized Poisson equations with millions of degrees of freedom arising in computational fluid dynamics applications. Unlike state-of-the-art learning approaches, our algorithm is capable of reducing the linear system residual to a given tolerance in a small number of iterations, independent of the problem size. Moreover, our method generalizes effectively to various systems beyond those encountered during training.

Kai Weixian Lan, Elias Gueidon, Ayano Kaneda et al.

We introduce a neural-preconditioned iterative solver for Poisson equations with mixed boundary conditions. Typical Poisson discretizations yield large, ill-conditioned linear systems. Iterative solvers can be effective for these problems, but only when equipped with powerful preconditioners. Unfortunately, effective preconditioners like multigrid require costly setup phases that must be re-executed every time domain shapes or boundary conditions change, forming a severe bottleneck for problems with evolving boundaries. In contrast, we present a neural preconditioner trained to efficiently approximate the inverse of the discrete Laplacian in the presence of such changes. Our approach generalizes to domain shapes, boundary conditions, and grid sizes outside the training set. The key to our preconditioner's success is a novel, lightweight neural network architecture featuring spatially varying convolution kernels and supporting fast inference. We demonstrate that our solver outperforms state-of-the-art methods like algebraic multigrid as well as recently proposed neural preconditioners on challenging test cases arising from incompressible fluid simulations.

Yongxu Jin, Dalton Omens, Zhenglin Geng et al.

Since loose-fitting clothing contains dynamic modes that have proven to be difficult to predict via neural networks, we first illustrate how to coarsely approximate these modes with a real-time numerical algorithm specifically designed to mimic the most important ballistic features of a classical numerical simulation. Although there is some flexibility in the choice of the numerical algorithm used as a proxy for full simulation, it is essential that the stability and accuracy be independent from any time step restriction or similar requirements in order to facilitate real-time performance. In order to reduce the number of degrees of freedom that require approximations to their dynamics, we simulate rigid frames and use skinning to reconstruct a rough approximation to a desirable mesh; as one might expect, neural-network-based skinning seems to perform better than linear blend skinning in this scenario. Improved high frequency deformations are subsequently added to the skinned mesh via a quasistatic neural network (QNN). In contrast to recurrent neural networks that require a plethora of training data in order to adequately generalize to new examples, QNNs perform well with significantly less training data.

Osman Akar, Yushan Han, Yizhou Chen et al.

We present learning-based implicit shape representations designed for real-time avatar collision queries arising in the simulation of clothing. Signed distance functions (SDFs) have been used for such queries for many years due to their computational efficiency. Recently deep neural networks have been used for implicit shape representations (DeepSDFs) due to their ability to represent multiple shapes with modest memory requirements compared to traditional representations over dense grids. However, the computational expense of DeepSDFs prevents their use in real-time clothing simulation applications. We design a learning-based representation of SDFs for human avatars whoes bodies change shape kinematically due to joint-based skinning. Rather than using a single DeepSDF for the entire avatar, we use a collection of extremely computationally efficient (shallow) neural networks that represent localized deformations arising from changes in body shape induced by the variation of a single joint. This requires a stitching process to combine each shallow SDF in the collection together into one SDF representing the signed closest distance to the boundary of the entire body. To achieve this we augment each shallow SDF with an additional output that resolves whether or not the individual shallow SDF value is referring to a closest point on the boundary of the body, or to a point on the interior of the body (but on the boundary of the individual shallow SDF). Our model is extremely fast and accurate and we demonstrate its applicability with real-time simulation of garments driven by animated characters.

Yongxu Jin, Yushan Han, Zhenglin Geng et al.

We present a novel paradigm for modeling certain types of dynamic simulation in real-time with the aid of neural networks. In order to significantly reduce the requirements on data (especially time-dependent data), as well as decrease generalization error, our approach utilizes a data-driven neural network only to capture quasistatic information (instead of dynamic or time-dependent information). Subsequently, we augment our quasistatic neural network (QNN) inference with a (real-time) dynamic simulation layer. Our key insight is that the dynamic modes lost when using a QNN approximation can be captured with a quite simple (and decoupled) zero-restlength spring model, which can be integrated analytically (as opposed to numerically) and thus has no time-step stability restrictions. Additionally, we demonstrate that the spring constitutive parameters can be robustly learned from a surprisingly small amount of dynamic simulation data. Although we illustrate the efficacy of our approach by considering soft-tissue dynamics on animated human bodies, the paradigm is extensible to many different simulation frameworks.