Research map / Probabilistic and scientific modeling / Differential equations and PDEs
Mosaic: A Benchmark Suite for Differentiable Physics Solvers
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Introduces Mosaic, a containerized benchmark suite that evaluates gradients of 14 differentiable PDE solvers across heat, structural, and fluid domains, measuring gradient accuracy, cost, conditioning, setup compatibility, and optimization convergence.
Problem
The practical cost of obtaining correct, usable gradients from differentiable PDE solvers is undocumented; integration effort, cost, gradient accuracy and conditioning vary widely across solvers and are discovered only by trial and error.
Approach
A uniform containerized (Tesseract) forward/VJP interface wraps heterogeneous solvers, and a standardized protocol with finite-difference references and solver-agnostic benchmark tasks compares them on gradient accuracy, cost, conditioning, compatibility and convergence.
Key results
- Across 14 solvers, VJP/forward cost and Jacobian conditioning vary by orders of magnitude; e.g. Warp-NS needs over 800 bespoke LOC for the initial-condition VJP while JAX-CFD and PhiFlow need 12 shared lines.
- All differentiable solvers reach cosine similarity >0.999 against central finite differences at their optimal perturbation size.
- On the F2 drag task only three of seven backends are structurally compatible; XLB and PICT reduce drag by around 60%, PhiFlow by about 35%.
Surprising findings
- Gradient accuracy is uniformly high across solvers, so practical barriers are memory, stability and setup compatibility rather than gradient correctness.
- Adam stagnates above 40% IC error on 3D NS recovery across all solvers, while L-BFGS with solenoidal projection drops it below 0.5%.
Limitations
- Evaluation relies on central finite differences as reference and out-of-the-box solver configurations, which may not reflect practitioner expertise.
- Benchmark tasks are deliberately simple standalone optimization problems without learned components, so hybrid pipeline failure modes are not captured.
- Results are pinned to specific solver versions and V100 hardware; relative rankings may differ on other hardware.
Experimental setup
Benchmarks: Mosaic
Metrics: cosine similarity relative l2 error wall-clock time peak VRAM Jacobian singular value spectrum IC error drag reduction convergence to optimum
Baselines: Adam L-BFGS OpenFOAM deal.II
Builds on: Tesseract JAX-CFD PhiFlow FEniCS Firedrake
Closest papers by meaning
- Learning a Neural Solver for Parametric PDEs to Enhance Physics-Informed Methods (ICLR 2025 · similarity 0.87)
- Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space (ICML 2025 · similarity 0.87)
- Gradient Alignment in Physics-informed Neural Networks: A Second-Order Optimization Perspective (NeurIPS 2025 · similarity 0.87)
- PRDP: Progressively Refined Differentiable Physics (ICLR 2025 · similarity 0.86)
- Metamizer: A Versatile Neural Optimizer for Fast and Accurate Physics Simulations (ICLR 2025 · similarity 0.86)
- Physics-informed Reduced Order Modeling of Time-dependent PDEs via Differentiable Solvers (NeurIPS 2025 · similarity 0.86)
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