David Millard

Ali Baheri, David Millard, Alireza Vahid

Distributed systems require fusing heterogeneous local probability distributions into a global summary over sparse and unreliable communication networks. Traditional consensus algorithms, which average distributions in Euclidean space, ignore their inherent geometric structure, leading to misleading results. Wasserstein barycenters offer a geometry-aware alternative by minimizing optimal transport costs, but their entropic approximations via the Sinkhorn algorithm typically require centralized coordination. This paper proposes a fully decentralized Sinkhorn algorithm that reformulates the centralized geometric mean as an arithmetic average in the log-domain, enabling approximation through local gossip protocols. Agents exchange log-messages with neighbors, interleaving consensus phases with local updates to mimic centralized iterations without a coordinator. To optimize bandwidth, we integrate event-triggered transmissions and b-bit quantization, providing tunable trade-offs between accuracy and communication while accommodating asynchrony and packet loss. Under mild assumptions, we prove convergence to a neighborhood of the centralized entropic barycenter, with bias linearly dependent on consensus tolerance, trigger threshold, and quantization error. Complexity scales near-linearly with network size. Simulations confirm near-centralized accuracy with significantly fewer messages, across various topologies and conditions.

David Millard, Ali Baheri

In robotics and multi-agent systems, fleets of autonomous agents often operate in subtly different environments while pursuing a common high-level objective. Directly pooling their data to learn a shared reward function is typically impractical due to differences in dynamics, privacy constraints, and limited communication bandwidth. This paper introduces an optimal transport-based approach to federated inverse reinforcement learning (IRL). Each client first performs lightweight Maximum Entropy IRL locally, adhering to its computational and privacy limitations. The resulting reward functions are then fused via a Wasserstein barycenter, which considers their underlying geometric structure. We further prove that this barycentric fusion yields a more faithful global reward estimate than conventional parameter averaging methods in federated learning. Overall, this work provides a principled and communication-efficient framework for deriving a shared reward that generalizes across heterogeneous agents and environments.

David Millard, Cecilia Alm, Rashid Ali et al.

Federated reinforcement learning typically aggregates value functions or policies by parameter averaging, which emphasizes expected return and can obscure statistical multimodality and tail behavior that matter in safety-critical settings. We formalize federated distributional reinforcement learning (FedDistRL), where clients parametrize quantile value function critics and federate these networks only. We also propose TR-FedDistRL, which builds a per client, risk-aware Wasserstein barycenter over a temporal buffer. This local barycenter provides a reference region to constrain the parameter averaged critic, ensuring necessary distributional information is not averaged out during the federation process. The distributional trust region is implemented as a shrink-squash step around this reference. Under fixed-policy evaluation, the feasibility map is nonexpansive and the update is contractive in a probe-set Wasserstein metric under evaluation. Experiments on a bandit, multi-agent gridworld, and continuous highway environment show reduced mean-smearing, improved safety proxies (catastrophe/accident rate), and lower critic/policy drift versus mean-oriented and non-federated baselines.

David Millard, Arielle Carr, Stéphane Gaudreault

Deep learning is revolutionizing weather forecasting, with new data-driven models achieving accuracy on par with operational physical models for medium-term predictions. However, these models often lack interpretability, making their underlying dynamics difficult to understand and explain. This paper proposes methodologies to estimate the Koopman operator, providing a linear representation of complex nonlinear dynamics to enhance the transparency of data-driven models. Despite its potential, applying the Koopman operator to large-scale problems, such as atmospheric modeling, remains challenging. This study aims to identify the limitations of existing methods, refine these models to overcome various bottlenecks, and introduce novel convolutional neural network architectures that capture simplified dynamics.

Carlos A. Pereira, Stéphane Gaudreault, Valentin Dallerit et al.

Recent machine-learning approaches to weather forecasting often employ a monolithic architecture, where distinct physical mechanisms (advection, transport), diffusion-like mixing, thermodynamic processes, and forcing are represented implicitly within a single large network. This representation is particularly problematic for advection, where long-range transport must be treated with expensive global interaction mechanisms or through deep, stacked convolutional layers. To mitigate this, we present PARADIS, a physics-inspired global weather prediction model that imposes inductive biases on network behavior through a functional decomposition into advection, diffusion, and reaction blocks acting on latent variables. We implement advection through a Neural Semi-Lagrangian operator that performs trajectory-based transport via differentiable interpolation on the sphere, enabling end-to-end learning of both the latent modes to be transported and their characteristic trajectories. Diffusion-like processes are modeled through depthwise-separable spatial mixing, while local source terms and vertical interactions are modeled via pointwise channel interactions, enabling operator-level physical structure. PARADIS provides state-of-the-art forecast skill at a fraction of the training cost. On ERA5-based benchmarks, the 1 degree PARADIS model, with a total training cost of less than a GPU month, meets or exceeds the performance of 0.25 degree traditional and machine-learning baselines, including the ECMWF HRES forecast and DeepMind's GraphCast.

Ali Baheri, David Millard, Ignacio Laguna Peralta

Fourier Neural Operators (FNOs) can greatly accelerate PDE simulation, but they are often used without formal guarantees that they preserve basic physical structure. We show that, once the trained weights and grid are fixed, the spectral convolution in an FNO is a linear map. As a result, the full forward pass is piecewise-linear and can be represented exactly in Z3's linear real arithmetic. We study two encodings. The exact encoding compiles the spectral convolution into a dense matrix multiplication, which is sound for both proofs and counterexamples. The lighter frozen encoding replaces the spectral path with a constant, making it faster but approximate. On 10 small FNO surrogates for 1D advection-diffusion-reaction (85 to 117 parameters, grids 8 to 32), the exact encoding gives 2 sound positivity proofs on linear (ReLU-free) models, 5 sound positivity counterexamples, and 10 sound mass-violation counterexamples; the remaining 3 positivity queries on ReLU models time out. For mass non-increase, Z3 finds worse counterexamples than both gradient-based falsification and Monte Carlo on 7 of 10 models. The frozen encoding scales to grid size 64 with sub-second positivity checks, but it no longer provides certificates for the original FNO. Overall, the results make the soundness--scalability tradeoff explicit and point to what is needed for formal verification of production-scale neural operators.

Anton Selitskiy, David Millard

This paper shows that the semi-dual formulation of the optimal transport problem has a degenerate saddle-point structure, and that its numerical solution is equivalent to solving a constrained optimization problem. We derive necessary and sufficient conditions for the convergence of Monge maps without requiring optimality of the dual potential. This analysis helps explain why, in practice, numerical algorithms often require more iterations to update the transport map than the potential.

David Millard, Arielle Carr, Stéphane Gaudreault et al.

We present PEARL (Preconditioner Enhancement through Actor-critic Reinforcement Learning), a novel approach to learning matrix preconditioners. Existing preconditioners such as Jacobi, Incomplete LU, and Algebraic Multigrid methods offer problem-specific advantages but rely heavily on hyperparameter tuning. Recent advances have explored using deep neural networks to learn preconditioners, though challenges such as misbehaved objective functions and costly training procedures remain. PEARL introduces a reinforcement learning approach for learning preconditioners, specifically, a contextual bandit formulation. The framework utilizes an actor-critic model, where the actor generates the incomplete Cholesky decomposition of preconditioners, and the critic evaluates them based on reward-specific feedback. To further guide the training, we design a dual-objective function, combining updates from the critic and condition number. PEARL contributes a generalizable preconditioner learning method, dynamic sparsity exploration, and cosine schedulers for improved stability and exploratory power. We compare our approach to traditional and neural preconditioners, demonstrating improved flexibility and iterative solving speed.

David Millard, Lars Lindemann, Ali Baheri

Uncertainty quantification for neural operators remains an open problem in the infinite-dimensional setting due to the lack of finite-sample coverage guarantees over functional outputs. While conformal prediction offers finite-sample guarantees in finite-dimensional spaces, it does not directly extend to function-valued outputs. Existing approaches (Gaussian processes, Bayesian neural networks, and quantile-based operators) require strong distributional assumptions or yield conservative coverage. This work extends split conformal prediction to function spaces following a two step method. We first establish finite-sample coverage guarantees in a finite-dimensional space using a discretization map in the output function space. Then these guarantees are lifted to the function-space by considering the asymptotic convergence as the discretization is refined. To characterize the effect of resolution, we decompose the conformal radius into discretization, calibration, and misspecification components. This decomposition motivates a regression-based correction to transfer calibration across resolutions. Additionally, we propose two diagnostic metrics (conformal ensemble score and internal agreement) to quantify forecast degradation in autoregressive settings. Empirical results show that our method maintains calibrated coverage with less variation under resolution shifts and achieves better coverage in super-resolution tasks.

David Millard, Arielle Carr, Stéphane Gaudreault et al.

We present DEF (\textbf{\ul{D}}iffusion-augmented \textbf{\ul{E}}nsemble \textbf{\ul{F}}orecasting), a novel approach for generating initial condition perturbations. Modern approaches to initial condition perturbations are primarily designed for numerical weather prediction (NWP) solvers, limiting their applicability in the rapidly growing field of machine learning for weather prediction. Consequently, stochastic models in this domain are often developed on a case-by-case basis. We demonstrate that a simple conditional diffusion model can (1) generate meaningful structured perturbations, (2) be applied iteratively, and (3) utilize a guidance term to intuitivey control the level of perturbation. This method enables the transformation of any deterministic neural forecasting system into a stochastic one. With our stochastic extended systems, we show that the model accumulates less error over long-term forecasts while producing meaningful forecast distributions. We validate our approach on the 5.625$^\circ$ ERA5 reanalysis dataset, which comprises atmospheric and surface variables over a discretized global grid, spanning from the 1960s to the present. On this dataset, our method demonstrates improved predictive performance along with reasonable spread estimates.

Eric Heiden, Christopher E. Denniston, David Millard et al.

To accurately reproduce measurements from the real world, simulators need to have an adequate model of the physical system and require the parameters of the model be identified. We address the latter problem of estimating parameters through a Bayesian inference approach that approximates a posterior distribution over simulation parameters given real sensor measurements. By extending the commonly used Gaussian likelihood model for trajectories via the multiple-shooting formulation, our chosen particle-based inference algorithm Stein Variational Gradient Descent is able to identify highly nonlinear, underactuated systems. We leverage GPU code generation and differentiable simulation to evaluate the likelihood and its gradient for many particles in parallel. Our algorithm infers non-parametric distributions over simulation parameters more accurately than comparable baselines and handles constraints over parameters efficiently through gradient-based optimization. We evaluate estimation performance on several physical experiments. On an underactuated mechanism where a 7-DOF robot arm excites an object with an unknown mass configuration, we demonstrate how our inference technique can identify symmetries between the parameters and provide highly accurate predictions. Project website: https://uscresl.github.io/prob-diff-sim

Eric Heiden, David Millard, Erwin Coumans et al.

Differentiable simulators provide an avenue for closing the sim-to-real gap by enabling the use of efficient, gradient-based optimization algorithms to find the simulation parameters that best fit the observed sensor readings. Nonetheless, these analytical models can only predict the dynamical behavior of systems for which they have been designed. In this work, we study the augmentation of a novel differentiable rigid-body physics engine via neural networks that is able to learn nonlinear relationships between dynamic quantities and can thus learn effects not accounted for in traditional simulators.Such augmentations require less data to train and generalize better compared to entirely data-driven models. Through extensive experiments, we demonstrate the ability of our hybrid simulator to learn complex dynamics involving frictional contacts from real data, as well as match known models of viscous friction, and present an approach for automatically discovering useful augmentations. We show that, besides benefiting dynamics modeling, inserting neural networks can accelerate model-based control architectures. We observe a ten-fold speed-up when replacing the QP solver inside a model-predictive gait controller for quadruped robots with a neural network, allowing us to significantly improve control delays as we demonstrate in real-hardware experiments. We publish code, additional results and videos from our experiments on our project webpage at https://sites.google.com/usc.edu/neuralsim.

Dominika Woszczyk, Stavros Petridis, David Millard

Speech recognition systems have improved dramatically over the last few years, however, their performance is significantly degraded for the cases of accented or impaired speech. This work explores domain adversarial neural networks (DANN) for speaker-independent speech recognition on the UAS dataset of dysarthric speech. The classification task on 10 spoken digits is performed using an end-to-end CNN taking raw audio as input. The results are compared to a speaker-adaptive (SA) model as well as speaker-dependent (SD) and multi-task learning models (MTL). The experiments conducted in this paper show that DANN achieves an absolute recognition rate of 74.91% and outperforms the baseline by 12.18%. Additionally, the DANN model achieves comparable results to the SA model's recognition rate of 77.65%. We also observe that when labelled dysarthric speech data is available DANN and MTL perform similarly, but when they are not DANN performs better than MTL.

Eric Heiden, David Millard, Erwin Coumans et al.

We present a differentiable simulation architecture for articulated rigid-body dynamics that enables the augmentation of analytical models with neural networks at any point of the computation. Through gradient-based optimization, identification of the simulation parameters and network weights is performed efficiently in preliminary experiments on a real-world dataset and in sim2sim transfer applications, while poor local optima are overcome through a random search approach.

David Millard, Eric Heiden, Shubham Agrawal et al.

A key ingredient to achieving intelligent behavior is physical understanding that equips robots with the ability to reason about the effects of their actions in a dynamic environment. Several methods have been proposed to learn dynamics models from data that inform model-based control algorithms. While such learning-based approaches can model locally observed behaviors, they fail to generalize to more complex dynamics and under long time horizons. In this work, we introduce a differentiable physics simulator for rigid body dynamics. Leveraging various techniques for differential equation integration and gradient calculation, we compare different methods for parameter estimation that allow us to infer the simulation parameters that are relevant to estimation and control of physical systems. In the context of trajectory optimization, we introduce a closed-loop model-predictive control algorithm that infers the simulation parameters through experience while achieving cost-minimizing performance.

Eric Heiden, David Millard, Hejia Zhang et al.

Intelligent agents need a physical understanding of the world to predict the impact of their actions in the future. While learning-based models of the environment dynamics have contributed to significant improvements in sample efficiency compared to model-free reinforcement learning algorithms, they typically fail to generalize to system states beyond the training data, while often grounding their predictions on non-interpretable latent variables. We introduce Interactive Differentiable Simulation (IDS), a differentiable physics engine, that allows for efficient, accurate inference of physical properties of rigid-body systems. Integrated into deep learning architectures, our model is able to accomplish system identification using visual input, leading to an interpretable model of the world whose parameters have physical meaning. We present experiments showing automatic task-based robot design and parameter estimation for nonlinear dynamical systems by automatically calculating gradients in IDS. When integrated into an adaptive model-predictive control algorithm, our approach exhibits orders of magnitude improvements in sample efficiency over model-free reinforcement learning algorithms on challenging nonlinear control domains.