differentiable nonlinear simulation

In engineering design, the non-differentiability of conventional CAE workflows—particularly mesh generation and physics simulation—hinders gradient-based high-dimensional optimization. To address this, we propose an end-to-end differentiable shape optimization framework: geometry is represented via signed distance fields (SDFs), and a 3D U-Net serves as a full-field surrogate model that directly learns the mapping from SDFs to physical fields (e.g., pressure, velocity), thereby bypassing non-differentiable components without requiring differentiable solvers or adjoint methods. The surrogate is embedded within a differentiable optimization pipeline, enabling backpropagation to compute gradients with respect to design parameters. Evaluated on aerodynamic shape optimization, our method achieves fully gradient-driven, efficient iterative design refinement. Results demonstrate substantial improvements in optimization efficiency and validate the framework’s feasibility and advantages in complex engineering applications.

Controlling deformation of soft objects is highly challenging due to strong nonlinearity and high-dimensional state spaces. This paper introduces the first end-to-end differentiable Material Point Method (MPM) simulator, enabling gradient-based optimization of control policies through full backpropagation. Our approach integrates an explicit hyperelastic constitutive model with a differentiable MPM discretization scheme, ensuring accurate computation of dynamics gradients. In the task of active damping for a hyperelastic rope, our framework achieves approximately 2× faster kinetic energy convergence, 20% lower steady-state energy, and only 3% of the computational cost compared to the MPPI baseline. The core contribution is the construction of the first fully differentiable MPM physics engine, rigorously validated for closed-loop control of complex deformable bodies—demonstrating both superior efficiency and accuracy in gradient-driven control synthesis.

This work addresses the longstanding absence of second-order derivatives (Hessians) for implicit functions in differentiable finite element physics. We present the first systematic derivation and implementation of an implicit Hessian-vector product algorithm based on primitive automatic differentiation operators, enabling PDE-constrained optimization. Our method leverages Jacobian-vector and vector-Jacobian products as core primitives, enabling seamless integration of Newton-CG and L-BFGS-B optimizers into finite element solvers. We establish a verifiable and scalable second-order implicit differentiation paradigm, thereby bridging a critical theoretical and practical gap in differentiable physics concerning Hessian computation. The approach is validated on four 2D/3D benchmark problems—spanning both linear and nonlinear regimes—with verified numerical accuracy. When combined with exact Hessians, Newton-CG accelerates convergence by 2–5× for traction identification and shape optimization tasks.

Existing physics engines for robotics struggle to simultaneously ensure stable simulation, high-fidelity rigid contact modeling, and full differentiability with respect to states, actions, and system parameters. To address this, we propose Dojo—the first end-to-end differentiable physics engine designed specifically for robotics. Dojo uniquely integrates variational integrators with a second-order cone nonlinear complementarity problem (NCP) solver, guaranteeing energy and momentum conservation during contact and enabling smooth, analytic gradient computation across contact events. It further employs a customized primal-dual interior-point method for efficient implicit differentiation. Evaluated on motion planning, policy optimization, and system identification tasks, Dojo demonstrates significantly improved gradient accuracy and faster optimization convergence in challenging rigid-contact scenarios.

This work addresses discrete-time nonlinear optimal control problems by unifying classical algorithms—including gradient descent, Gauss–Newton, Newton’s method, and differential dynamic programming (DDP)—within a differentiable programming framework. Methodologically, it introduces the first modular, end-to-end differentiable algorithm template library built upon linear/quadratic approximations (e.g., LQR), enabled by automatic differentiation. Theoretically, it provides a unified derivation of computational complexity and sufficient optimality conditions across all methods. Practically, it incorporates adaptive line search and regularization strategies, and validates efficacy on benchmark tasks such as autonomous racing with a bicycle model. All implementations are open-sourced, demonstrating both efficient gradient propagation and strong generalization across diverse control problems.

This work addresses the high memory overhead and neglect of local structure in gradient computation for implicit nonlinear solvers within differentiable simulation. The authors propose a solver-level differentiation method that constructs an adjoint algorithm symmetric to the forward solve by reverse-scanning a block-structured implicit solver, entirely avoiding the assembly of a global Jacobian matrix. For the first time, adjoint computation is aligned with the block structure of the forward solver, combining vertex-block descent with reverse-colored Gauss–Seidel sweeps to enable efficient backpropagation using only local 3×3 adjoint solves. This approach leverages operator-view approximations of the inverse and its transpose. On a single GPU, it achieves a 33× speedup and 71× reduction in memory compared to unrolled automatic differentiation, enabling, for the first time, differentiable elastic dynamics simulation of million-contact coupled soft bodies with up to 8 million vertices.

This study addresses the PDE-constrained inverse problem of correcting Reynolds-averaged Navier–Stokes (RANS) turbulence models by introducing an end-to-end differentiable framework. The approach embeds a PDE solver into PyTorch’s automatic differentiation graph via implicit layers and incorporates a trainable additive correction term, enabling joint optimization of model parameters or neural networks. Built upon the BROADCAST solver, this work presents a unified and user-friendly differentiable PDE interface and, for the first time, achieves end-to-end differentiable correction for compressible-flow RANS models. The framework successfully optimizes the production-term coefficient in the Spalart–Allmaras model and reconstructs the eddy viscosity field in two canonical cases—NASA’s wall-mounted hump and the VKI LS-59 turbine blade—demonstrating its effectiveness and flexibility for turbulence modeling and broader physics-informed PDE inverse problems.

This work addresses gradient distortion in existing differentiable simulators under frictional contact and large deformations, which stems from mathematical inconsistencies and leads to optimization failure. We propose the first unified, fully GPU-accelerated differentiable simulator that enables stable, high-fidelity gradient computation across contact states. Our approach integrates a rigorously Markovian position-velocity manifold coupling, a mass-aligned preconditioner, a soft Fischer–Burmeister friction operator, and a finite element singularity resolution technique. We further introduce a long-horizon consistency mechanism and a unified contact stability strategy, enabling—for the first time—mathematically rigorous modeling of both frictional contact and hyperelastic materials within a differentiable framework. The resulting low-noise, high-fidelity gradients significantly narrow the Sim-to-Real gap and enhance the reliability of physical system identification and control in tasks such as dexterous manipulation and cloth folding.

This work addresses the challenge of system identification bias in sim-to-real transfer for soft robotics, which arises from constitutive model misspecification or sparse observations—particularly pronounced when geometric morphology serves as a design variable. To mitigate this, the authors propose the Residual Acceleration Field Learning (RAFL) framework, which augments a base simulator with a local residual dynamics field that is independent of mesh topology and discretization. RAFL leverages locally shared features and is trained end-to-end, enabling zero-shot generalization across morphologies. By integrating differentiable simulation with sparsely labeled real-world observations, RAFL consistently outperforms conventional system identification approaches in both sim-to-sim and sim-to-real experiments, effectively avoiding negative transfer and supporting cumulative improvements in simulation fidelity throughout continuous optimization processes.

This work addresses the challenges in parameter learning for differential-algebraic equation (DAE) systems with state-dependent events, where algebraic variables are implicitly defined, event times depend on parameters, and reset maps introduce discontinuities. The authors propose a differentiable parameter optimization framework that formulates the problem as a constrained least-squares problem incorporating DAE dynamics, algebraic constraints, guard conditions, and reset maps. Two novel gradient computation methods are developed: a piecewise differentiable calculus based on automatic differentiation and an explicit discrete adjoint method. Notably, the study clarifies— for the first time—that residual terms in the adjoint approach arise from equality constraints rather than heuristic penalties. Under the assumptions of fixed event ordering and transversal guard crossings, both methods yield valid gradients along forward simulation trajectories, demonstrating the correctness and applicability of the proposed framework.