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Post-training language models with continuous physics rewards lets one policy write accurate multi-PDE solvers, and a smaller trained model can beat prompting a much larger frontier model.

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2026-07-14 11:25 UTC pith:B23UHDYW

load-bearing objection Solid multi-PDE RL post-training paper: continuous physics rewards beat binary validity and can make a small coder model beat frontier prompting on in-distribution solver generation; transfer is real but selective. the 3 major comments →

arxiv 2607.10474 v1 pith:B23UHDYW submitted 2026-07-11 cs.LG cs.AIcs.CE

Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards

Pengfei Cai , Utkarsh Utkarsh , Alan Edelman , Christopher Vincent Rackauckas , Rafael Gomez-Bombarelli This is my paper
classification cs.LG cs.AIcs.CE
keywords reinforcement learningPDE solverscode generationverifiable rewardslarge language modelsscientific computingpost-trainingnumerical methods
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

Building a reliable numerical PDE solver still requires expert choices about discretizations, stability limits, and boundary handling—choices that can make two programs that both run differ by orders of magnitude in solution quality. This paper claims that those graded scientific preferences can be written into a language model’s parameters, not only recovered at inference time with prompting and debugging loops. The method, RLVP, first warms up a model on a bank of teacher solvers, then applies reinforcement learning with a hybrid verifier: hard checks require the code to execute and return finite, correctly shaped fields, while continuous rewards score trajectory accuracy against hidden references and residual consistency with the PDE. One policy trained jointly across eight equation families improves pass rates and solution error over base and supervised baselines, shows selective zero-shot gains on held-out PDEs by recombining numerical motifs, and allows a smaller post-trained model to outperform direct prompting of a frontier model on in-distribution solver generation. If the claim holds, scientific code agents can amortize numerical reliability into the model rather than relying only on larger static models or per-problem scaffolding.

Core claim

The paper establishes that Reinforcement Learning with Verifiable Physics (RLVP)—post-training a single language-model policy with a hybrid binary–continuous verifier that multiplies program validity by function-space trajectory accuracy and PDE-residual consistency—substantially improves multi-PDE solver code generation. Across model scales, RLVP raises pass@1 and pass@8 and drives median best-of-8 nRMSE to about 6×10^{-4} on eight seen families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems; a smaller RLVP-trained model outperforms prompting a frontier model under the same direct-generation protocol; and the policy shows selective zero-shot transfer to held-out P

What carries the argument

The hybrid RLVP reward R = V · R_traj · R_phys. Validity V is a hard gate (executes, correct shape, finite values). Continuous factors are expected pass probabilities under exponential tolerances on normalized space-time L2 trajectory error and reference-relative residual error. After supervised fine-tuning on a multi-PDE solver bank, GRPO uses group-relative advantages from this reward to shift probability toward programs that are both executable and physically accurate.

Load-bearing premise

The method assumes that continuous scores against hidden numerical reference trajectories and residuals, with fixed exponential temperatures and a fixed success error threshold, measure true physical correctness rather than closeness to particular reference solvers, grids, and operator combinations.

What would settle it

Run the same GRPO setup with validity-only reward versus full continuous physics reward under matched seeds and rollouts: if validity-only matches or beats full RLVP on median best-of-8 nRMSE and held-out transfer, continuous physics is not the driver; or re-evaluate RLVP solvers on new grids, parameters, and independent reference solvers outside the paper’s suite—if high training scores collapse while validity stays high, the signal is overfit to the hidden references.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • A single post-trained policy can serve multiple PDE families instead of specializing one model per equation.
  • Continuous physical accuracy rewards improve solution quality and cross-PDE transfer beyond binary execution feedback alone.
  • Smaller verifier-trained models can outperform larger static frontier models on in-distribution PDE solver generation under direct generation.
  • Trained policies recombine numerical motifs (stencils, CFL substepping, RK schemes, boundary handling) into solvers for related unseen PDEs.
  • Future solver-writing systems should put more weight on parameter updates from verifiable physics feedback, not only on inference-time prompting and debugging.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same hybrid continuous-verifier pattern may transfer to other scientific code tasks that admit graded residual or conservation-error signals, such as molecular-dynamics integrators or finite-element assembly.
  • If motif recombination is the transfer mechanism, expanding the solver bank with more operator types could widen zero-shot coverage without a proportional increase in training PDE families.
  • Reference-relative rewards may under-reward stylistically different solvers that still satisfy the PDE better than the hidden reference on new instances.
  • Combining RLVP with retrieval, debugging loops, or verifier-guided search—settings the paper leaves for future work—could compound the amortized gains isolated under direct generation.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper introduces RLVP, a GRPO-based post-training method that adapts LLMs to generate numerical PDE solver code using a hybrid verifier: binary execution validity gates multiplied by continuous rewards for function-space trajectory error (R_traj) and reference-relative residual consistency (R_phys). After SFT on a multi-scheme solver bank spanning eight PDE families (hyperbolic, parabolic, elliptic, incompressible flow), a single policy is trained jointly and evaluated by pass@k and best-of-k nRMSE against hidden numerical references. Across 3B/7B/14B Qwen2.5-Coder models, RLVP substantially improves over base and SFT checkpoints on seen PDEs, can outperform static prompting of larger frontier models under the same direct-generation protocol, and yields selective gains on held-out PDEs that recombine trained numerical motifs. An ablation shows that adding R_traj improves accuracy beyond validity-only RL.

Significance. If the results hold under broader scrutiny, this is a clear and timely contribution: it moves scientific code generation from inference-time scaffolding to amortized post-training with graded physical feedback, and it is among the first multi-PDE RLVR pipelines with continuous rather than binary verifiers. Strengths include a carefully documented solver bank with self-convergence and MMS checks, controlled Base/SFT/RLVP comparisons at three scales, frontier baselines under a matched protocol, a validity-only ablation, and an intended open release of the verifier environment and recipes. The practical claim that a smaller post-trained model can beat frontier prompting on in-distribution solver generation is well supported and of real interest to scientific computing and LLM post-training communities.

major comments (3)
  1. §3.5 Eqs. (3)–(5) and Appendix A fix T_traj=0.05, T_phys=2.0, and the success threshold nRMSE≤10^{-2} without sensitivity analysis. These free parameters define both the dense training signal and the pass@k metric. Because the central claim is that continuous physics rewards (not just validity) drive gains, the paper should report at least a limited sweep or robustness check on T_traj (and preferably T_phys and the success threshold) for the 7B model on seen and held-out aggregates; otherwise it is hard to know how much of Table 1 and Figs. 8–9 depends on this particular temperature/threshold choice.
  2. §4.3 and Appendix F present compositionality and zero-shot transfer as recombination of stencils, CFL control, and time-stepping motifs. The quantitative gains on held-out PDEs are real, but the strongest cases (heat ⊂ reaction–diffusion; 2D advection as a 1D lift; ARD as advection+reaction+diffusion) sit inside the trained operator algebra, while KdV, Darcy–reaction, and advection–diffusion remain weak (Table 16). The abstract and contribution list should more tightly match this selectivity. Please either (i) add a controlled motif-ablation or feature-frequency analysis that separates recombination from generic code quality, or (ii) explicitly frame transfer as selective numerical-motif reuse rather than broad physics internalization.
  3. §3.5 Eq. (1) and Eq. (4): L_traj is measured only against the environment’s fixed numerical references, and R_phys is reference-relative. This is a legitimate RLVR design, but it leaves open whether the policy is learning scheme-agnostic physical correctness or matching reference-solver idiosyncrasies (numerical diffusion, projection residuals, CFL patterns). A load-bearing check for the “verifiable physics” claim would be evaluation under a changed reference family, grid family, or residual discretization not used in training—at least for one or two PDEs—or an explicit limitation stating that rewards are reference-anchored and transfer is within a related numerical-methods neighborhood.
minor comments (6)
  1. Figure 2 and Table 1: clarify whether Claude Sonnet 4.5 and other API baselines used identical system/user prompt constraints (NumPy/SciPy only, fixed interface) as the local models; any prompt mismatch would weaken the “smaller model beats frontier” comparison.
  2. §4.2: Incompressible Navier–Stokes is the clear capacity bottleneck (only 14B RLVP succeeds strongly). A short discussion of why projection structure is harder to learn from the current bank would help readers set expectations.
  3. Related Work: CodePDE and SciML-agent systems are cited; a short explicit contrast on compute amortization (post-training once vs. per-query refinement) would sharpen positioning without new experiments.
  4. Appendix C’s exponential-tolerance derivation is useful; cross-reference it more clearly from §3.5 so readers see that R_traj is an expected-pass probability, not an ad-hoc soft score.
  5. Typos/clarity: “DeepSeek V4 Flash” labeling in Table 1 should be double-checked against the actual API model name; ensure consistent PDE naming (e.g., reaction_diffusion1d vs. Reaction Diffusion 1D) between tables and figures.
  6. Figure 11 (solver component frequencies) is informative; reporting sample sizes and a simple statistical comparison (RLVP vs SFT) would make the motif-reuse narrative less anecdotal.

Circularity Check

0 steps flagged

No load-bearing circular derivation: RLVP optimizes execution-grounded rewards against hidden numerical references and evaluates on held-out cases/PDEs; standard RLVR, not prediction-by-construction.

full rationale

This paper’s central chain is methodological and empirical, not a first-principles derivation that reduces to its inputs. SFT warm-starts on a curated multi-PDE solver bank; GRPO then updates a policy using R(y,c)=V·R_traj·R_phys (Eq. 5), where V is hard executability and R_traj/R_phys are continuous scores from function-space error and reference-relative residual consistency on hidden instances (Eqs. 1–4; §3.5). Evaluation uses fixed test cases and 10 held-out PDE families never used as SFT targets or RL prompts (§4.1, §4.3). Training on a metric and reporting the same metric on disjoint cases is ordinary RLVR, not self-definitional circularity: the policy is not forced to reproduce a fitted scalar or a quantity defined from the evaluation labels. Appendix C’s exponential-tolerance reading of R_traj is a re-interpretation of the soft reward as an expected binary pass probability, not a claim that a prediction equals a fit. Author self-citations appear only in related SciML context (e.g., physics-constrained generative models) and do not underwrite uniqueness or force the RLVP design. The skeptic concern that dense rewards may overfit reference-solver idiosyncrasies or operator-recombination structure is a generalization/correctness risk, not a circular reduction (no Eq. X ≡ Eq. Y by construction). Mild residual risk only: rewards and success thresholds are defined relative to the same class of numerical references used for grading, so reported nRMSE gains partly measure what was optimized—but held-out PDEs and parameter/IC splits keep this from being tautological. Score 1 reflects that mild coupling without any exhibited circular step.

Axiom & Free-Parameter Ledger

5 free parameters · 5 axioms · 2 invented entities

The central empirical claim rests on standard RL and numerical-analysis background plus several design choices that are free parameters or paper-specific constructions: exponential tolerance temperatures, success threshold, the multiplicative hybrid reward, and the assumption that hidden numerical references define graded physical correctness for training. No new physical entities are postulated; the invented pieces are methodological (RLVP verifier and multi-PDE solver-bank recipe).

free parameters (5)
  • T_traj (trajectory reward temperature)
    Sets the scale of the exponential expected-pass reward R_traj=exp(-nRMSE/T_traj); fixed at 0.05 without reported sensitivity analysis, yet directly shapes ranking of valid solvers.
  • T_phys (residual reward temperature)
    Temperature for residual-consistency factor; fixed at 2.0 and multiplies into the complete reward, affecting how much residual mismatch attenuates trajectory reward.
  • success_nrmse_threshold
    Binary success for pass@k uses nRMSE≤0.01 for all PDEs; this threshold converts continuous error into the headline reliability metrics.
  • KL coefficient β and GRPO/LoRA optimization hyperparameters
    β=0.001, actor LR 2e-6, LoRA rank 32, 8 rollouts, 50 epochs, etc., are hand-chosen training knobs that the reported gains depend on.
  • Prompt-form sampling weights (0.5/0.35/0.15)
    Mixture of generic, parameter-conditioned, and IC-conditioned prompts is a design choice that shapes the learned policy distribution.
axioms (5)
  • domain assumption Group-relative policy optimization with token-mean advantages and KL anchor to an SFT reference is a valid post-training method for code-generating policies.
    Invoked throughout §3.6; taken from DeepSeekMath/GRPO literature without re-derivation.
  • domain assumption Normalized L2 trajectory error against a hidden numerical reference is a sufficient proxy for physical solver quality across the studied PDE families.
    Defines R_traj via Eq. (1) and §3.5; standard in PDEBench-style evaluation but still an assumption about what 'correctness' means.
  • ad hoc to paper An exponential distribution over acceptance tolerances yields an appropriate dense RLVR-style pass probability for continuous diagnostics.
    Appendix C motivates this as maximum-entropy on R≥0; other tolerance laws are left to future work, so the reward shape is a modeling choice.
  • domain assumption Teacher solvers in the curated bank are numerically adequate warm-start targets (self-convergence/MMS/cross-checks suffice).
    Appendix B.3 verification protocol underpins the SFT prior; exact closed-form checks are unavailable for many nonlinear cases.
  • standard math Standard finite-difference/volume/spectral numerical analysis results used in the solver bank are reliable background.
    Citations to LeVeque, Toro, Chorin, Trefethen, etc., supply the teacher schemes.
invented entities (2)
  • RLVP hybrid verifier (V · R_traj · R_phys) no independent evidence
    purpose: Gate invalid programs and rank valid PDE solver code by continuous function-space accuracy and residual consistency during RL post-training.
    Core methodological construct of the paper; not a physical entity, but a new reward object the claims depend on.
  • Multi-PDE solver-bank + joint post-training recipe across eight families no independent evidence
    purpose: Provide a shared executable prior and a single policy trained over hyperbolic, parabolic, elliptic, and incompressible-flow systems.
    Dataset/system contribution enabling the multi-PDE claims; evidence is internal experimental, not external prior validation of the bank as a community standard.

pith-pipeline@v1.1.0-grok45 · 39176 in / 4051 out tokens · 59840 ms · 2026-07-14T11:25:37.463523+00:00 · methodology

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Cite this review

Pith. "Pith review of Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards." pith.science (2026). https://pith.science/paper/B23UHDYW

@misc{pith2026260710474,
  author       = {Pith},
  title        = {Pith review of: Reinforcement Learning with Verifiable Physics: Post-training LLMs with Continuous Rewards},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/B23UHDYW}},
  note         = {Machine review of arXiv:2607.10474}
}
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read the original abstract

Partial differential equations (PDEs) are foundational to modeling in science and engineering, but constructing reliable numerical solvers remains labor-intensive, demanding expert knowledge of discretization schemes, stability conditions, and boundary treatments. Recent work has begun to frame PDE solving as a code-generation task for large language models (LLMs), yet existing approaches operate primarily at inference time: relying on prompting, debugging, self-refinement, and test-time scaling rather than adapting the model itself. In parallel, reinforcement learning with verifiable rewards has emerged as a post-training paradigm for code and math reasoning, but its verifiers are typically binary: a compiler runs, or a test passes. Such signals discard the graded structure of scientific correctness, where two solvers may both execute and yet differ in solution accuracy by orders of magnitude. In this work, we introduce RLVP: Reinforcement Learning with Verifiable Physics, an RL post-training framework for multi-PDE solver code generation. RLVP addresses this verifiability gap with a hybrid verifier: hard program-validity checks ensure executability, while continuous physics rewards score function-space accuracy and PDE-residual consistency. A single policy is post-trained across diverse PDE families spanning hyperbolic, parabolic, elliptic, and incompressible-flow systems. RLVP improves over both pre-trained and supervised-only baselines on PDE benchmarks, and shows zero-shot improvement transfer to held-out PDEs. We show that a smaller LLM post-trained with RLVP can outperform prompting a frontier model on in-distribution PDE solver generation. The trained policy shows evidence of compositionality in numerical motifs: it recombines stencils, time-stepping schemes, and boundary-handling primitives learned from the PDEs used in training into generated solvers for unseen PDE problems.

Figures

Figures reproduced from arXiv: 2607.10474 by Alan Edelman, Christopher Vincent Rackauckas, Pengfei Cai, Rafael Gomez-Bombarelli, Utkarsh Utkarsh.

Figure 1
Figure 1. Figure 1: Overview of RLVP for multi-PDE solver code generation. We warm-start the policy with supervised fine-tuning (SFT) on a multi-PDE solver bank across diverse PDE problems. During RL, sampled solver programs are executed by the RLVP verifier and scored by a reward combining validity, physical accuracy against hidden numerical references, and physics residual consistency. Group-relative advantages update the p… view at source ↗
Figure 2
Figure 2. Figure 2: Progress of evaluation metrics during post-training. We compare Base (Qwen2.5-Coder￾Instruct), SFT, and RLVP post-trained checkpoints on the eight seen PDE tasks. Dashed line: Claude Sonnet 4.5 baseline evaluated with the same k = 8 sampling protocol. Evaluation problems and metrics. We evaluate all models on fixed PDE prompts and case sets. The seen-PDE evaluation uses the same eight PDE families as train… view at source ↗
Figure 7
Figure 7. Figure 7 [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 3
Figure 3. Figure 3: Comparison across local and frontier models on seen PDEs. Base, SFT, RLVP, and frontier LLMs are compared by valid execution rate, pass@1, pass@8, and median best-of-8 nRMSE. With RLVP, parameter updates from executable physical feedback can make a smaller policy more likely to generate numerically reliable solver programs than prompting a much larger static model. Our multi-PDE setup enables all models to… view at source ↗
Figure 4
Figure 4. Figure 4: Evaluation on 10 held-out PDE tasks. We compare Base, SFT, and RLVP checkpoints on unseen PDE prompts not used during SFT or RL training [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Compositional transfer to unseen advection–reaction–diffusion (ARD) (RLVP 7B model). Left: reference and best-of-8 generated solutions for the held-out ARD problem with β = 1, ν = 0.02, and ρ = 2. Right: numerical scheme evidences from generated PDE solver snippets: a Burgers solver from RLVP, and ARD solvers from SFT and RLVP post-trained models. Evidence of compositional numerical methods transfer [PITH… view at source ↗
Figure 6
Figure 6. Figure 6: Self-convergence of the solver bank across the 8 PDE families. Each panel shows the [PITH_FULL_IMAGE:figures/full_fig_p023_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Training reward over RL steps. RLVP training reward increases smoothly across 3B, 7B, and 14B Qwen2.5-Coder-Instruct models. The curves show that verifier feedback provides a stable optimization signal after the SFT warm start. 25 [PITH_FULL_IMAGE:figures/full_fig_p025_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Ablation of physical accuracy reward vs. validity-only reward for 8 seen PDE tasks. We compare validity-only RL, R = V , against validity plus physical accuracy, R = V Rtraj, for the 7B model. Here, Rtraj is computed from function-space nRMSE against hidden references. Validity-only RL attains a similar valid execution rate, but adding Rtraj improves pass@1 (0.71 vs. 0.58), pass@8 (0.84 vs. 0.76), and medi… view at source ↗
Figure 9
Figure 9. Figure 9: Ablation of physical accuracy reward vs. validity-only reward for 10 unseen PDE tasks. On the 10 held-out PDE prompts, adding Rtraj to validity improves valid execution rate (0.45 vs. 0.29), pass@1 (0.10 vs. 0.03), pass@8 (0.33 vs. 0.21), and median best-of-8 nRMSE (1.3 × 10−1 vs. 1.8 × 10−1 ) relative to validity-only RL. The physical accuracy reward is therefore important for cross-PDE transfer [PITH_FU… view at source ↗
Figure 10
Figure 10. Figure 10: Progress of pass@8 evaluation metrics during post-training. Same setup as [PITH_FULL_IMAGE:figures/full_fig_p026_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Solver component frequencies across model checkpoints. We parse generated solver code for numerical-method features. For solver generation on held-out PDE problems, RLVP post￾trained checkpoints tend to more frequently use periodic boundary logic, np.roll-based stencils, CFL/substep control, and Runge–Kutta time stepping. These features align with the successful compositional transfer examples in [PITH_F… view at source ↗
Figure 12
Figure 12. Figure 12: Dimension lift transfer: 1D advection to 2D advection. The matched seen problem gives the 1D advection solver generated by the RLVP post-trained model. The held-out RLVP solver lifts the transport structure to two dimensions using x/y derivative stencils, directional CFL control with dx and dy, and RK4 time stepping, reaching median nRMSE 5.349 × 10−3 . The SFT model generated solver is unstable, with med… view at source ↗
Figure 13
Figure 13. Figure 13: Higher-order synthesis: Cahn–Hilliard. The held-out Cahn–Hilliard PDE prompt requires a fourth-order structure. The solver generated by the RLVP post-trained model constructs the chemical potential µ = −ϵ 2uxx + u 3 − u and applies a second Laplacian, reaching median nRMSE 2.616 × 10−4 compared with 2.144 × 10−2 for the SFT finetuned model. This suggests that the post-trained model could adapt familiar fi… view at source ↗
Figure 14
Figure 14. Figure 14: Diffusion-subset transfer: reaction–diffusion to heat. Heat is the diffusion-only subset of the seen reaction–diffusion family during SFT and RL. The solver generated by RLVP post-trained model drops the reaction source while retaining a local periodic Laplacian and explicit substepping, improving median best-of-8 nRMSE from 2.07 × 10−1 for SFT to 8.11 × 10−5 after RL. 28 [PITH_FULL_IMAGE:figures/full_fi… view at source ↗
Figure 15
Figure 15. Figure 15: Generated solver from the RLVP post-trained model for 1D viscous Burgers. Best selected solver from the 7B RLVP checkpoint on the seen viscous Burgers task. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Generated solver from the RLVP post-trained model for 1D advection. Best selected solver from the 7B RLVP checkpoint on the seen 1D advection task. 31 [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Generated solver from the RLVP post-trained model for 1D reaction–diffusion. Best selected solver from the 7B RLVP checkpoint on the seen reaction–diffusion task. 32 [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Generated solver from the RLVP post-trained model for 1D diffusion–sorption. Best selected solver from the 7B RLVP checkpoint on the seen diffusion–sorption task. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Generated solver from the RLVP post-trained model for 2D reaction–diffusion. Best selected solver from the 7B RLVP checkpoint on the seen 2D reaction–diffusion task. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Generated solver from the RLVP post-trained model for 2D Darcy flow. Best selected solver from the 7B RLVP checkpoint on the seen Darcy flow task. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Generated solver from the RLVP post-trained model for 2D shallow water equations. Best selected solver from the 7B RLVP checkpoint on the seen shallow water task. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Generated solver from the RLVP post-trained model for 2D incompressible Navier– Stokes. Best selected solver from the 7B RLVP checkpoint on the seen incompressible Navier–Stokes task. 37 [PITH_FULL_IMAGE:figures/full_fig_p037_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Burgers 1D solutions example for the steeper-multimode initial condition with [PITH_FULL_IMAGE:figures/full_fig_p038_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: Diffusion–sorption 1D solutions example for the uniform initial condition with [PITH_FULL_IMAGE:figures/full_fig_p038_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: Reaction–diffusion 1D solutions example for the front-like initial condition with [PITH_FULL_IMAGE:figures/full_fig_p038_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: Darcy 2D solutions example for the channel diffusion field with [PITH_FULL_IMAGE:figures/full_fig_p038_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Advection–reaction–diffusion 1D solutions (unseen PDE) example for the front-like initial [PITH_FULL_IMAGE:figures/full_fig_p038_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Heat 1D solutions (unseen PDE) example for the single-sine initial condition with [PITH_FULL_IMAGE:figures/full_fig_p039_28.png] view at source ↗
Figure 29
Figure 29. Figure 29: Allen–Cahn 2D solutions (unseen PDE) example for the mixed-waves initial condition [PITH_FULL_IMAGE:figures/full_fig_p040_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Example generic prompt used in the 1D linear advection task. [PITH_FULL_IMAGE:figures/full_fig_p041_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Example parameter-conditioned prompt used in the 1D reaction–diffusion task. [PITH_FULL_IMAGE:figures/full_fig_p042_31.png] view at source ↗
Figure 32
Figure 32. Figure 32: Example parameter and initial condition conditioned prompt used in the 2D Darcy flow [PITH_FULL_IMAGE:figures/full_fig_p042_32.png] view at source ↗

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