Per Astronomix ad Astra: High-Order Differentiable (Magneto)hydrodynamics with Energy-Conserving Self-Gravity

Leonard Storcks; Nils Thuerey; Tobias Buck

REVIEW 3 major objections 6 minor 296 references

Per Astronomix ad Astra: High-Order Differentiable (Magneto)hydrodynamics with Energy-Conserving Self-Gravity

T0 review · 3 major / 6 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A new differentiable simulation framework couples a fifth-order finite-difference (magneto)hydrodynamics scheme to an energy-conserving self-gravity source term, making gradient-based astrophysical inverse modeling practical.

desk verdict A substantial, honest engineering-plus-methods paper on differentiable MHD+self-gravity; the flagship self-gravity order claim is observed rather than proven, but the paper deserves a serious referee. read the letter →

arxiv 2607.18176 v1 pith:FJTYZQJC submitted 2026-07-20 astro-ph.IM physics.flu-dyn

Leonard Storcks , Nils Thuerey , Tobias Buck This is my paper · ORCID

classification astro-ph.IMphysics.flu-dyn

keywords magnetohydrodynamicsself-gravitydifferentiablesimulatorautomaticdifferentiationWENOfinite-differenceschemeconstrainedtransportenergyconservationinversemodeling

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper presents a GPU-accelerated simulator for astrophysical (magneto)hydrodynamics built on automatic differentiation, arguing that it combines three properties that have been difficult to hold simultaneously: fifth-order spatial accuracy, energy conservation in the presence of self-gravity at the semi-discrete level, and exact end-to-end gradients. The central claim is that a high-order finite-difference scheme can couple to gravity through a gravitational energy flux constructed from the same numerical mass flux used by the hydrodynamics, corrected by an explicit deconvolution term, so the discrete gravitational energy telescopes exactly while remaining fourth-order accurate. If correct, this makes field-level inverse modeling of magnetized, self-gravitating flows practical; the paper demonstrates optimizing a turbulent 128-cubed initial velocity field with roughly six million parameters against a target image. The claim is supported by convergence tests, energy errors below 1e-8 in a gravitational collapse, gradient verification against analytical functional derivatives and finite differences, and single-GPU runtimes of the same order as a widely used performance-portable reference code.

What carries the argument

The central identity is the box-filter/deconvolution relation of finite-difference flux-form schemes: for f'(xi) ≈ (f̂_{i+1/2} − f̂_{i−1/2})/Δx to be high-order, the numerical face flux f̂ must be a high-order face value of the deconvolved flux f̃ = f − (Δx²/24)f'' + O(Δx⁴). The new self-gravity scheme constructs the gravitational energy flux from the WENO mass flux and adds an explicit O(Δx²) correction so the face flux of the product Φf is deconvolved to fourth order while retaining exact telescoping — the property that yields semi-discrete energy conservation at full spatial order.

What would settle it

Run the large-amplitude slab advection at N = 16 to N = 128 with the correction term enabled and disabled and compare convergence slopes: if the corrected scheme does not hold at least fourth-order (ideally fifth, as reported) while the uncorrected one stalls at second order, the deconvolution identity is not doing the claimed work. Alternatively, in a mild Evrard collapse with the positivity limiter inactive, verify that the total-energy error scales as O(Δt⁴); deviations would contradict the semi-discrete conservation argument.

Watch

Extended reading notes

Core claim

The core discovery: self-gravity can couple to a high-order finite-difference (M)HD scheme without breaking energy conservation. The numerical mass flux is a face value of a deconvolved flux, so the natural energy source −∇·(ρvΦ) − Φ∂tρ telescopes exactly but stays second-order, because deconvolution does not commute with multiplication. An explicit correction from second-order cell-centered derivatives averaged to faces restores fourth-order accuracy while preserving exact telescoping. The corrected scheme shows fifth-order convergence on a large-amplitude slab test and total-energy errors below 1e-8 in a gravitational collapse. The same framework is fully differentiable, with gradients val

Load-bearing premise

The load-bearing premise is that, in smooth regions, the WENO interface mass flux is a high-order face value of the deconvolved flux, so the second-order-derived correction term restores fourth-order accuracy without breaking exact telescoping; the paper observes, rather than proves, the resulting fifth-order convergence, and the argument explicitly degrades at discontinuities.

Editorial extensions

If this is right

Field-level inverse modeling over millions of parameters becomes practical: the paper recovers a turbulent 128³ initial velocity field (~6.3 million degrees of freedom) by matching a target column-density image.
The corrected flux-based self-gravity coupling cuts total-energy error in a gravitational collapse from up to 18% to below 1e-8 at 128³ cells, with residuals scaling as O(Δt⁴) under the Runge-Kutta integrator.
On smooth problems, the fifth-order scheme reaches the same error as second- and third-order reference schemes roughly three to four orders of magnitude faster for a given runtime on a single GPU.
Gradients through the full nonlinear solver are certified: reverse-mode automatic differentiation converges at the nominal spatial order to analytic functional derivatives on smooth flows, and to one-sided finite differences through a shock tube.
New stability measures — positivity-preserving flux blending and deep-void protection — keep the high-order scheme stable in high-Mach, strongly magnetized turbulence at resolutions where previous implementations broke.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The deconvolution-correction construction should transfer to other nonlinear source terms in finite-difference schemes — optically thin cooling, cosmic-ray pressure, radiation pressure — wherever a product of a face flux and a cell-centered field must stay high-order without breaking conservation.
A clean falsifiable consequence of the semi-discrete argument: over many dynamical times in a smooth, mildly collapsing run, total energy should remain pinned at the time-integration error floor rather than drifting, cleanly separating the corrected scheme from simple source-term coupling even at low resolution.
Because the Jacobian-vector product is matrix-free from a single automatic-differentiation pass, the eigenmode/stability-analysis demonstration generalizes to routine diagnostics on any statistically stationary astrophysical flow, not just idealized shear layers.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Hi — this is a substantial paper. It delivers a JAX differentiable (M)HD code with fifth-order finite-difference WENO/CT following Seo & Ryu, a novel fourth-order energy-conserving self-gravity coupling, a careful AD verification suite, and scaling results to 16 GPUs. If the central claims hold, it gives the field a credible tool for field-level inference and solver-in-the-loop problems with MHD+gravity. That's a real capability increase, not an incremental one.

What's new: the flux-based self-gravity source term with the deconvolution correction (Eq. 86), the analytic-functional-derivative gradient checks (Sec. 5.1), and the eigenmode initialization via AD Jacobian diagonalization (Sec. 6.1). The MHD core is a reimplementation of Seo & Ryu, and the FV baseline follows Pang & Wu; that's clearly stated and cited. Credit where due: the paper is unusually transparent — it explicitly notes the Jeans test masks the uncorrected scheme, the multi-shooting terminal loss is measured on a lifted/infeasible trajectory, and the order claim in Sec. 3.4.4 relies on the 'evidently carries a small enough constant' observation.

The soft spot is the order guarantee for the self-gravity scheme. The correction is derived to O(Δx^4) under smoothness, but the substitution f → f_hat is only valid to O(Δx^2), and the only direct evidence is a single slab-advection test. The stress-test concern — that in genuinely 3D, non-aligned flows the fourth-order error constant could be non-negligible, degrading the scheme to effectively 2nd/3rd order — is a real risk, not manufactured. However, the authors do not overclaim: they report observed fifth-order convergence on the slab test, not a proof. This is exactly what a referee should push on. A 3D off-axis convergence test and/or a formal error bound would strengthen the paper considerably.

I don't see a circularity problem. The energy conservation is a designed semi-discrete identity, tested against external benchmarks and analytic waves; the free parameters listed are not fitted to the claims. The exact artifact isn't version-pinned (code, Pallas kernels, environment), which is a reproducibility concern for a methods paper but is fixable.

Bottom line: this deserves a serious referee. I'd suggest engaging with it — for the differentiable-simulator audience and anyone doing gradient-based astro inference, this is a useful reference and a good benchmark. I would read it and, if I were in that line of work, cite it.

Referee Report

3 major / 6 minor

Summary. The paper presents astronomix, a JAX/Python differentiable ideal-(M)HD simulator with optional self-gravity. Its central technical contributions are: (i) a fifth-order finite-difference WENO constrained-transport MHD scheme following Seo & Ryu (2023), augmented by positivity/stability measures, a low-storage Runge-Kutta option, and an agent-generated Pallas GPU backend; (ii) a new self-gravity coupling that solves the Poisson equation with FFTs and constructs the gravitational energy source from the numerical mass flux so that the discrete gravitational-energy change telescopes exactly, yielding semi-discrete (time-continuous) energy conservation, with an explicit O(Δx^4) deconvolution correction intended to restore spatial order; (iii) demonstrations that reverse-mode automatic differentiation through the solver is accurate and scalable, including analytic-functional-derivative and finite-difference validation, eigenmode initialization of a Kelvin-Helmholtz instability, multiple-shooting optimization, and field-level inference over roughly 6.3 million parameters; and (iv) performance and scaling comparisons against AthenaPK. The forward solver is tested on standard hydro/MHD problems, turbulence, and Evrard collapse, and the paper is unusually candid about several limitations of the self-gravity tests and time-integration behavior.

Significance. If the claims hold, this is a significant contribution: a differentiable, GPU-scalable, high-order MHD+self-gravity code would enable gradient-based inference, sensitivity analysis, and solver-in-the-loop machine learning in astrophysics. The paper has real strengths: the energy-conservation identity is derived and tested, AD gradients are checked against hand-derived functional derivatives, the Evrard collapse shows energy errors below roughly 10^-8 for the flux-based schemes in the smooth regime, and the open-source code plus generated Pallas backend are validated against the JAX reference. Several limitations are explicitly acknowledged, including the inability of the Jeans-wave test to separate the schemes and the first-order temporal energy-error behavior when the positivity limiter engages. The main risk is the spatial-order guarantee of the self-gravity correction, which currently rests on an asymptotic formula and one smooth slab test; this is the principal reason for my recommendation.

major comments (3)

[Sec. 3.4.2–3.4.4, Eqs. (84)–(86), Fig. 13] The flagship claim of a fourth-order self-gravity scheme is not established at the same level as the energy-conservation identity. Equation (86) is an asymptotic correction derived under the smoothness assumption, stated in Eq. (85), that the WENO interface mass flux is a high-order face value of the deconvolved flux \tilde f. The only direct numerical evidence is the slab advection test (Sec. 3.4.4), a single configuration with wave vector and advection velocity parallel to the box diagonal. The text itself says the remaining fourth-order error 'evidently carries a small enough constant' (Sec. 3.4.4); that is an empirical observation, not an order bound. In genuinely three-dimensional non-aligned flows the O(Δx^4) constant is uncontrolled, and if it is not small the scheme could be effectively second- or third-order, undercutting the abstract's 'fourth-order self-gravity scheme'. I ask
[Sec. 3.4.5, Figs. 14–17] The advertised O(Δt^4) energy convergence of the flux-based schemes is conditional on the positivity-preserving limiter not engaging. The text states that the cold Evrard collapse at 32^3 crashes unless the PP limiter is enabled, and that where the limiter is active it reduces the temporal convergence of the energy error to first order, because the limited flux becomes an explicit function of Δt and breaks the RK4 cancellation. The O(Δt^4) behavior is demonstrated only for the milder e_th,0=0.2 case (Fig. 16). Since the limiter is necessary in exactly the gravitational-collapse regime where self-gravity matters, the practical discrete energy-conservation order in realistic use is first order in time, not fourth. This does not invalidate the semi-discrete identity, but the paper should either provide a limiter formulation that preserves high-order RK cancellation or state prominently that
[Sec. 3.4.3, Fig. 12] The Jeans linear-wave test is explicitly unable to separate the corrected from the uncorrected flux-based scheme: the correction term is second order in the perturbation amplitude and falls below round-off at ε=10^-6. Consequently the entire spatial-order evidence for the corrected scheme rests on the slab test alone (Fig. 13), which covers only N=16–96 and one wavenumber/amplitude. In view of the central role of the order claim, I would like to see at least one additional independent test—for example, a manufactured solution with a non-trivial 3D potential and velocity field, or a small-amplitude Jeans wave at larger ε but still in the linear regime—before the fourth-order statement is accepted as general.

minor comments (6)

[Sec. 3.4.4] Replace the phrase 'the remaining fourth-order error evidently carries a small enough constant' with a quantitative statement, e.g., the measured leading constant or an additional test that controls that constant. As written, this sentence is the only support for the observed fifth-order convergence.
[Sec. 4.1, Fig. 18] The runtime comparison uses different CFL numbers and integrators (FD at CFL=1.5; AthenaPK at 0.3/0.4). The conclusion is supported by the error-vs-runtime panel, but the text should state explicitly that the total-runtime comparison combines spatial order, integrator efficiency, and CFL choice; the time-per-iteration panel already shows a 1.4–4.8× penalty.
[Sec. 3.1.1, Eq. (18)] The WENO ε=10^-7 is dimensional and 'assumes code units of order one'. A short note on how ε should be rescaled when the code is used with astrophysical unit systems would be useful, since all WENO weights depend on it.
[Sec. 5.1 / Sec. 6.3] The million-parameter field-level inference demo (Sec. 6.3) is qualitative and uses no regularization, as the authors note. For reproducibility, please report the optimizer schedule, number of iterations, checkpoint count, and wall-clock cost of the 128^3 logo inversion.
[Sec. 3.4.2] In the implementation of Eq. (86), the correction is evaluated with cell-centered point values f_i=(ρv_x)_i and then averaged to faces, while the derivation writes \hat f and \hat f'. The text explains why this is asymptotically valid (f = \hat f + O(Δ^2)), but a reader may be confused; please state explicitly that the final O(Δ^4) error is unaffected by this replacement.
[Sec. 3.1.2] The constrained-transport construction is formally fourth-order in the transverse interpolations, and fifth-order convergence is inherited from the WENO truncation error, as in Seo & Ryu (2023). The paper should make clear that this is an observed, not proven, property of the composite scheme.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: central derivations are self-contained and independently validated.

full rationale

Astronomix's flagship claims do not reduce to fitted inputs or self-citations. The self-gravity energy-conserving source term is explicitly constructed to satisfy the discrete identity sum_i V_i S_E,i = -∂_t E_grav,d (Eq. 72); the paper acknowledges this by-design property, so the reported energy conservation in Evrard's collapse is an implementation check of a constructed invariant, not a prediction derived from the same invariant. The claimed fourth-order accuracy of the correction (Eq. 86) follows from an explicit asymptotic deconvolution argument (Eqs. 81–85) with no fitted constants, and the order claim is tested against independent analytical solutions (Jeans waves, slab advection) and external references (Hanawa & Mullen 2025). The differentiability claims are validated against hand-derived functional derivatives derived independently from the linearized Euler equations (Appendix F), not from the discretization itself. Benchmarks against AthenaPK and standard shock tests are external. Self-citations (Storcks & Buck 2024, 2025) provide context for radially-symmetric solvers and solver-in-the-loop applications but are not load-bearing for the central numerical or differentiability results. No uniqueness theorem is imported, and no ansatz is smuggled through self-citation. The paper candidly states the smoothness limitation of its order argument in Sec. 3.4.2, which is an acknowledged scope restriction, not circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central numerical claims do not require fitting any constants to data: the energy-conserving source term is constructed to satisfy a discrete identity and is then tested externally. The free numbers listed are standard WENO/stability/setup parameters. The main hidden dependencies are smoothness of the flow for the deconvolution argument, boundary assumptions for energy conservation, and the spectral Poisson solve.

free parameters (4)

WENO smoothness-indicator epsilon = 1e-7
Hand-set dimensional floor in Eq. (18) to avoid division by zero; standard but not derived from data.
Deep-void blend factor beta = 8
Chosen for the ISM turbulence stabilization in Sec. 3.3.9; hand-set, not derived.
Turbulence forcing amplitude F0 = 3.5
OU-driving amplitude in Eq. (63) chosen so that vrms is approximately 1; setup-specific, not fitted to the central energy-conservation claim.
Turbulence density floor and velocity cap = rho_min=0.02, v_max=50
Stabilization choices for the high-Mach ISM runs, adopted from Seo & Ryu's Fortran code; hand-set.

assumptions (5)

domain assumption WENO interface fluxes coincide with high-order face values of the deconvolved flux in smooth regions (Eqs. 84-85).
Needed to justify the self-gravity correction; the argument breaks at discontinuities, so the high-order/energy-conservation claim is restricted to smooth flows.
domain assumption The surface term in the gravitational energy balance vanishes (Eq. 69): density falls off sufficiently fast or there is no mass flux through the boundary.
Used in the continuous and discrete energy-conservation argument; holds for the periodic/isolated test problems but is not generally true for arbitrary domains.
domain assumption A Fourier/FFT Poisson solve with Jeans swindle or Hockney-Eastwood represents self-gravity accurately enough for the finite-difference coupling.
The hydro scheme is finite-difference while gravity uses spectral Green's functions; the accuracy matching is asserted rather than formally proven.
standard math Explicit Runge-Kutta integration of the quadratic gravitational energy produces the claimed O(Delta t^4) fully discrete energy error.
Used to explain why semi-discrete conservation is not exact in the discrete simulation; standard for quadratic invariants but stated, not proved in detail.
domain assumption The ideal MHD equations with adiabatic or isothermal closure are the correct physical model (Eqs. 1-5).
The simulator inherits the standard ideal-MHD model, including the mu0=1 convention and the constrained-transport requirement div B = 0.

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Cite this review

Pith. "Pith review of Per Astronomix ad Astra: High-Order Differentiable (Magneto)hydrodynamics with Energy-Conserving Self-Gravity." pith.science (2026). https://pith.science/paper/FJTYZQJC

@misc{pith2026260718176,
  author       = {Pith},
  title        = {Pith review of: Per Astronomix ad Astra: High-Order Differentiable (Magneto)hydrodynamics with Energy-Conserving Self-Gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FJTYZQJC}},
  note         = {Machine review of arXiv:2607.18176}
}

abstract

We present astronomix, a performant differentiable (magneto)hydrodynamics simulator written in Python/JAX. We demonstrate how automatic differentiation, validated against hand-derived analytical functional derivatives and finite differences, enables inverse modeling over millions of parameters and allows for sensitivity and stability analysis as well as correct eigenmode initialization. The differentiability of astronomix furthermore enables training machine-learning models inside the simulator. On a single GPU at a given resolution, astronomix has runtimes of the same order of magnitude as the GPU-optimized code AthenaPK but reaches far lower errors on smooth problems due to its higher order. astronomix scales to multiple GPUs ($\sim 6.5$ strong scaling speedup on $8$ GPUs) and multiple nodes ($\sim 76\%$ weak scaling efficiency on $16$ GPUs over $4$ nodes). We also present a novel fourth-order self-gravity scheme which complements the fifth-order finite difference constrained transport magnetohydrodynamics scheme implemented in astronomix. To maximize performance, we created an agentic skill that generates and validates custom Pallas GPU kernels from our JAX reference code and test suite. The simulator is available at https://github.com/leo1200/astronomix.

Figures

Figures reproduced from arXiv: 2607.18176 by the authors.

**Figure 1.** Convergence of the 3D linear sound-wave test: 𝐿1 error of the solution versus linear resolution 𝑁 (grid (2𝑁, 𝑁, 𝑁)) at fixed integration time, for the finite-volume scheme (FV) and the finite-difference scheme on the native-JAX and Pallas backends (FD, the two backends overlap exactly). Dashed and dotted lines mark 𝑂(𝑁−2) and 𝑂(𝑁−5) reference slopes. 𝜌(𝑥) = { 1.0, if 𝑥 < 0.5 0.125, otherwise 𝑢(𝑥) = 0 𝑝(𝑥) = { 1.0, i… view at source ↗

**Figure 2.** Sod shock tube: density, velocity and pressure at 𝑡 = 0.2 for the finite-volume method with HLLC Riemann solver and Minmod and Superbee limiters, and the finite-difference scheme, all at 𝑁 = 200 cells, compared with the analytical Riemann solution. The inset zooms in on the contact discontinuity, where the Superbee limiter is least diffusive but slightly overshoots. 3.3.3. 1D Double blast Next we consider the double… view at source ↗

**Figure 3.** Double blast wave of Woodward and Colella, 1984: density at 𝑡 = 0.038 for the finite-volume method with HLL and HLLC Riemann solvers (Minmod limiter) and the finite-difference scheme, all at 𝑁 = 400 cells, together with a converged 𝑁 = 10000 finite-volume reference (solid line). The finite-difference result is closest to the reference at the central density peak. In [PITH_FULL_IMAGE:figures/full_fig_p017_3.png] view at source ↗

Figures from the paper (30 more)

**Figure 4.** Figure 4: Sedov–Taylor blast wave at 𝑡 = 0.1: radial profiles of density, velocity magnitude and pressure (columns) for the finite-volume method with Minmod limiter and HLL, HLLC and AM-HLLC Riemann solvers and the finite-difference method (rows), all at 2563 cells. Grey points …

**Figure 5.** Figure 5: Convergence of the 3D circularly polarized Alfvén-wave test (double precision): average 𝐿1 error versus linear resolution 𝑁 (grid (2𝑁, 𝑁, 𝑁)) for the finite-volume (FV) and finite-difference (FD) schemes and the AthenaPK reference code in a second-order (second-order v…

**Figure 6.** Figure 6: For the Orszag-Tang vortex with initial conditions given in Eq. 58 the density field (left, finite difference at 10242 cells) and a cut through it at 𝑦 = 0.625𝜋 (right) at 𝑡 = 3.0 are shown. The cut compares the finite-volume and finite-difference schemes at 2002 cells…

**Figure 7.** Figure 7: Strongly magnetized 3D MHD blast wave at 𝑡 = 0.02 and 2563 cells. The four left panels show central slices of the finite-difference solution (density, kinetic energy 𝑣 2 , magnetic pressure 𝐵 2 and thermal pressure); the right column shows |𝐵| 2 and pressure along the …

**Figure 8.** Figure 8: Central density slice of the MHD blast wave at 1283 (top) and 2563 (bottom) cells for the finite-volume Lax and HLL schemes (one with a second-order implicit midpoint magnetic step and one with a first-order implicit Euler magnetic step) and the finite-difference schem…

**Figure 9.** Figure 9: Magnetically driven jet at 𝑡 = 5, density slice through the jet axis, computed with the finite-difference scheme at 10243 cells. of the fluid state, the realization is exactly reproducible for a given timestep sequence and contributes trivially to the tangent/adjoint, …

**Figure 10.** Figure 10: Driven isothermal-MHD turbulence: mid-plane slices of the magnetic energy log10 𝐸𝐵 for the supersonic ISM regime (𝑀turb ≈ 10, 𝛽𝑝 = 0.1; top) and the subsonic ICM regime (𝑀turb ≈ 0.5, 𝛽𝑝 = 106 ; bottom), each at 2563 (left) and 5123 (right) cells, computed with the fin…

**Figure 11.** Figure 11: Saturated power spectra of the driven isothermal-MHD turbulence, time-averaged over the stationary phase: density 𝑃𝜌 (𝑘) (left), kinetic 𝐸𝐾 (𝑘) (middle), and magnetic 𝐸𝐵 (𝑘) (right) for the ISM (top) and ICM (bottom) regimes at 2563 and 5123 cells. Dashed lines mark t…

**Figure 12.** Figure 12: Convergence of the 𝐿1 density error in the Jeans linear wave test for the three self-gravity coupling schemes. The near-identical high-order convergence of the uncorrected flux-based scheme here is a small-amplitude artifact (its correction term is second order in the…

**Figure 13.** Figure 13: Convergence of the 𝐿1 density error in the slab advection test for the three self-gravity coupling schemes. embedded in a tenuous ambient medium. The gas is initially at rest and nearly cold, with a uniform specific internal energy of 𝑒𝑡ℎ = 0.05 (in units of 𝐺𝑀∕𝑅, wit…

**Figure 14.** Figure 14: Evolution of the energy components and of the total energy error in Evrard’s collapse for the different self-gravity coupling schemes (1283 cells, double precision). For a milder collapse with a warmer massive sphere with 𝑒𝑡ℎ,0 = 0.2 we demonstrate the convergence of …

**Figure 15.** Figure 15: Radial profiles in Evrard’s collapse at 𝑡 = 0.8 for the different self-gravity coupling schemes (1283 cells, double precision) [PITH_FULL_IMAGE:figures/full_fig_p036_15.png]

**Figure 16.** Figure 16: Convergence of the total energy error with the time step for the mild version of Evrard’s collapse (𝑒𝑡ℎ,0 = 0.2). 4.1. Single-GPU runtimes and memory usage [PITH_FULL_IMAGE:figures/full_fig_p036_16.png]

**Figure 17.** Figure 17: Convergence of the total energy error with spatial resolution (under a CFL time step) for the mild version of Evrard’s collapse (𝑒𝑡ℎ,0 = 0.2). comparable to that of RK3+PPM (≈ 91 s [≈ 58 s]). Due to its fifth-order convergence, the FD (Pallas) scheme reaches far lower…

**Figure 18.** Figure 18: Single-GPU performance on the 3D circularly polarized Alfvén-wave test (double precision), comparing FV (JAX), FD (Pallas) and AthenaPK in a second-order (VL2+PLM) and third-order (RK3+PPM) setup, run on an NVIDIA A100 in double precision. Top: average 𝐿1 error versus…

**Figure 19.** Figure 19: Single-GPU memory for the 3D sound-wave (hydro) test versus resolution 𝑁 (grid 2𝑁 ×𝑁 ×𝑁): total compiled memory (left) and transient-to-argument memory ratio (right). FD (Pallas) is leanest, with a transient/argument ratio close to 2 [PITH_FULL_IMAGE:figures/full_fig…

**Figure 20.** Figure 20: As [PITH_FULL_IMAGE:figures/full_fig_p039_20.png]

**Figure 21.** Figure 21: Strong-scaling speedup 𝑇 (1)∕𝑇 (𝐺) versus resolution for the FD (Pallas) solver: hydro on 4 GPUs (H100) and hydro/MHD on 8 GPUs (H200). Baselines exceeding single-GPU memory use a power-law extrapolation [PITH_FULL_IMAGE:figures/full_fig_p040_21.png]

**Figure 22.** Figure 22: Reverse-mode AD gradient of the final-state 𝐿2 cost with respect to the full initial density field, for a 3D Gaussian density perturbation (𝑁 = 64, evolved until it has separated into outgoing acoustic waves), compared against the exact analytic (Fourier-space) functi…

**Figure 23.** Figure 23: Convergence of the reverse-mode AD gradient to the exact analytic (Fourier-space) functional derivative of the linearized acoustic problem, as a function of grid resolution 𝑁. The finite-difference solver converges at its nominal fifth order (𝑁−5), the finite-volume s…

**Figure 24.** Figure 24: Shock-tube AD verification: the kink-immune min-one-sided finite-difference residual ‖ min𝐹 ,𝐵 |𝑔AD − 𝑔FD|‖2∕‖𝑔AD‖2 as a function of the FD step ℎ. It tracks the one-sided truncation ∝ ℎ to ∼ 7 ⋅ 10−6; the JAX and Pallas finite-difference backends overlap. of recomput…

**Figure 25.** Figure 25: Reverse-mode performance versus the number of checkpoints 𝑐 for 𝑁steps = 481 adaptive steps: backward-pass runtime (left), theoretical minimal recomputation count (middle) and compiled backward-pass memory (right). The Pallas finite-difference adjoint backend based on…

**Figure 26.** Figure 26: Inverse modeling landscape. The choice of inference method depends on the number of predicted parameters 𝑑 and the number of inferences on new observations 𝑁. The upfront training cost of amortized simulation-based inference (SBI) methods increases with parameter spac…

**Figure 27.** Figure 27: Eigenvalue spectra of the AD-linearized solver Jacobian for a single 𝑥-Fourier mode, for the homogeneous upper and lower states and for the sharp and smoothed shear interfaces. Points are colored by the effective transverse wavenumber 𝑘 ef f 𝑦 ∕𝑘𝑦,Nyq. The homogeneous…

**Figure 28.** Figure 28: Growth of the mode amplitude ln(𝐴(𝑡)∕𝐴0 ) versus 𝑡∕𝜏KH. The eigenmode initialization (blue) follows the predicted growth rate Re(𝜆) (dotted) from 𝑡 = 0, while the pure velocity perturbation (red dashed) shows an initialization transient before joining the growing bran…

**Figure 29.** Figure 29: Final states at 𝑡 = 4.04 ≃ 1.5 𝜏KH for the pure velocity initialization (top) and the eigenmode initialization (bottom), showing log density, transverse velocity 𝑣𝑦 , and pressure. The eigenmode case, free of the initial transient, reaches a more developed billow stag…

**Figure 30.** Figure 30: Illustration of PDE-constrained optimization paradigms along the full-space vs reduced-space discussion. In single shooting, the PDE-constraint is fully enforced by the simulator while in full-space methods like PINNs and ODIL the PDE-constraint is also enforced by th…

**Figure 31.** Figure 31: Reconstruction for one successful initialization per horizon (recovered ic_er r < 0.1): single shooting at 𝑇 = 20 𝑡 𝑔 (top two rows, ic_er r = 0.001) and multiple shooting at 𝑇 = 60 𝑡 𝑔 (bottom two rows, ic_er r = 0.046). Columns show the truth, the cold-start initial…

**Figure 32.** Figure 32: Single vs. multiple shooting on the KH instability with an Adam optimizer, 𝑁 = 256, 𝑀 = 8, 16 cold initializations per configuration. Rows: short horizon 𝑇 = 20 𝑡 𝑔 (top, single shooting wins) and long horizon 𝑇 = 60 𝑡 𝑔 (bottom, multiple shooting wins). Columns: term…

**Figure 33.** Figure 33: Field-level inference with the differentiable astronomix MHD solver. The initial velocity field of a driven 1283 turbulence box is optimized so that its 𝑧-axis density projection forms the astronomix logo after ∼ one turbulent crossing time. Panels show volume renderi…

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