ICLR 2023 Papers

Neural networks can be trained to solve partial differential equations (PDEs) by using the PDE residual as the loss function. This strategy is called "physics-informed neural networks" (PINNs), but it currently cannot produce high-accuracy solutions, typically attaining about $0.1\%$ relative error. We present an adversarial approach that overcomes this limitation, which we call competitive PINNs (CPINNs). CPINNs train a discriminator that is rewarded for predicting mistakes the PINN makes. The discriminator and PINN participate in a zero-sum game with the exact PDE solution as an optimal strategy. This approach avoids squaring the large condition numbers of PDE discretizations, which is the likely reason for failures of previous attempts to decrease PINN errors even on benign problems. Numerical experiments on a Poisson problem show that CPINNs achieve errors four orders of magnitude smaller than the best-performing PINN. We observe relative errors on the order of single-precision accuracy, consistently decreasing with each epoch. To the authors' knowledge, this is the first time this level of accuracy and convergence behavior has been achieved. Additional experiments on the nonlinear Schr{\"o}dinger, Burgers', and Allen--Cahn equation show that the benefits of CPINNs are not limited to linear problems.

Dynamical systems are found in innumerable forms across the physical and biological sciences, yet all these systems fall naturally into equivalence classes: conservative or dissipative, stable or unstable, compressible or incompressible. Predicting these classes from data remains an essential open challenge in computational physics on which existing time-series classification methods struggle. Here, we propose, phase2vec, an embedding method that learns high-quality, physically-meaningful representations of low-dimensional dynamical systems without supervision. Our embeddings are produced by a convolutional backbone that extracts geometric features from flow data and minimizes a physically-informed vector field reconstruction loss. The trained architecture can not only predict the equations of unseen data, but also produces embeddings that encode meaningful physical properties of input data (e.g. stability of fixed points, conservation of energy, and the incompressibility of flows) more faithfully than standard blackbox classifiers and state-of-the-art time series classification techniques. We additionally apply our embeddings to the analysis of meteorological data, showing we can detect climatically meaningful features. Collectively, our results demonstrate the viability of embedding approaches for the discovery of dynamical features in physical systems.

Neural networks with sinusoidal activations have been proposed as an alternative to networks with traditional activation functions. Despite their promise, particularly for learning implicit models, their training behavior is not yet fully understood, leading to a number of empirical design choices that are not well justified. In this work, we first propose a simplified version of such sinusoidal neural networks, which allows both for easier practical implementation and simpler theoretical analysis. We then analyze the behavior of these networks from the neural tangent kernel perspective and demonstrate that their kernel approximates a low-pass filter with an adjustable bandwidth. Finally, we utilize these insights to inform the sinusoidal network initialization, optimizing their performance for each of a series of tasks, including learning implicit models and solving differential equations.

Zeroth-order (ZO) optimization, in which the derivative is unavailable, has recently succeeded in many important machine learning applications. Existing algorithms rely on finite difference (FD) methods for derivative estimation and gradient descent (GD)-based approaches for optimization. However, these algorithms suffer from query inefficiency because many additional function queries are required for derivative estimation in their every GD update, which typically hinders their deployment in real-world applications where every function query is expensive. To this end, we propose a trajectory-informed derivative estimation method which only employs the optimization trajectory (i.e., the history of function queries during optimization) and hence can eliminate the need for additional function queries to estimate a derivative. Moreover, based on our derivative estimation, we propose the technique of dynamic virtual updates, which allows us to reliably perform multiple steps of GD updates without reapplying derivative estimation. Based on these two contributions, we introduce the zeroth-order optimization with trajectory-informed derivative estimation (ZoRD) algorithm for query-efficient ZO optimization. We theoretically demonstrate that our trajectory-informed derivative estimation and our ZoRD algorithm improve over existing approaches, which is then supported by our real-world experiments such as black-box adversarial attack, non-differentiable metric optimization, and derivative-free reinforcement learning.

Deep learning based approaches like Physics-informed neural networks (PINNs) and DeepONets have shown promise on solving PDE constrained optimization (PDECO) problems. However, existing methods are insufficient to handle those PDE constraints that have a complicated or nonlinear dependency on optimization targets. In this paper, we present a novel bi-level optimization framework to resolve the challenge by decoupling the optimization of the targets and constraints. For the inner loop optimization, we adopt PINNs to solve the PDE constraints only. For the outer loop, we design a novel method by using Broyden's method based on the Implicit Function Theorem (IFT), which is efficient and accurate for approximating hypergradients. We further present theoretical explanations and error analysis of the hypergradients computation. Extensive experiments on multiple large-scale and nonlinear PDE constrained optimization problems demonstrate that our method achieves state-of-the-art results compared with strong baselines.

Boundary conditions (BCs) are important groups of physics-enforced constraints that are necessary for solutions of Partial Differential Equations (PDEs) to satisfy at specific spatial locations. These constraints carry important physical meaning, and guarantee the existence and the uniqueness of the PDE solution. Current neural-network based approaches that aim to solve PDEs rely only on training data to help the model learn BCs implicitly, however, there is no guarantee of BC satisfaction by these models during evaluation. In this work, we propose Boundary enforcing Operator Network (BOON) that enables the BC satisfaction of neural operators by making structural changes to the operator kernel. We provide our refinement procedure, and demonstrate the satisfaction of physics-based BCs such as Dirichlet, Neumann, and periodic by the solutions obtained by BOON. Numerical experiments based on multiple PDEs with a wide variety of applications indicate that the proposed approach ensures satisfaction of BCs, and leads to more accurate solutions over the whole domain. The proposed method exhibits a (2X-20X) improvement in accuracy (0.000084 relative $L^2$ error for Burgers' equation). Code available at: https://github.com/amazon-science/boon.

Neural Motion Planners (NMPs) have emerged as a promising tool for solving robot navigation tasks in complex environments. However, these methods often require expert data for learning, which limits their application to scenarios where data generation is time-consuming. Recent developments have also led to physics-informed deep neural models capable of representing complex dynamical Partial Differential Equations (PDEs). Inspired by these developments, we propose Neural Time Fields (NTFields) for robot motion planning in cluttered scenarios. Our framework represents a wave propagation model generating continuous arrival time to find path solutions informed by a nonlinear first-order PDE called Eikonal Equation. We evaluate our method in various cluttered 3D environments, including the Gibson dataset, and demonstrate its ability to solve motion planning problems for 4-DOF and 6-DOF robot manipulators where the traditional grid-based Eikonal planners often face the curse of dimensionality. Furthermore, the results show that our method exhibits high success rates and significantly lower computational times than the state-of-the-art methods, including NMPs that require training data from classical planners.

Motion mimicking is a foundational task in physics-based character animation. However, most existing motion mimicking methods are built upon reinforcement learning (RL) and suffer from heavy reward engineering, high variance, and slow convergence with hard explorations. Specifically, they usually take tens of hours or even days of training to mimic a simple motion sequence, resulting in poor scalability. In this work, we leverage differentiable physics simulators (DPS) and propose an efficient motion mimicking method dubbed $\textbf{DiffMimic}$. Our key insight is that DPS casts a complex policy learning task to a much simpler state matching problem. In particular, DPS learns a stable policy by analytical gradients with ground-truth physical priors hence leading to significantly faster and stabler convergence than RL-based methods. Moreover, to escape from local optima, we utilize an \textit{Demonstration Replay} mechanism to enable stable gradient backpropagation in a long horizon. Extensive experiments on standard benchmarks show that DiffMimic has a better sample efficiency and time efficiency than existing methods (e.g., DeepMimic). Notably, DiffMimic allows a physically simulated character to learn back-flip after 10 minutes of training and be able to cycle it after 3 hours of training, while DeepMimic requires about a day of training to cycle back-flip. More importantly, we hope DiffMimic can benefit more differentiable animation systems with techniques like differentiable clothes simulation in future research. Our code is available at https://github.com/diffmimic/diffmimic. Qualitative results can be viewed at https://diffmimic-demo-main-g7h0i8.streamlitapp.com.