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arxiv: 2606.02623 · v1 · pith:4WDZTI7Enew · submitted 2026-05-29 · 💻 cs.NE · cs.AI· cs.LG

Oscillatory State-Space Models as Inductive Biases for Physics-Informed Neural PDE Solvers

Abhishek Chandra , Taniya Kapoor This is my paper

Pith reviewed 2026-06-28 20:08 UTC · model grok-4.3

classification 💻 cs.NE cs.AIcs.LG
keywords physics-informed neural networksstate-space modelsoscillatory dynamicsPDE solversspectral methodsinductive biasestime-dependent PDEshigh-dimensional problems
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The pith

Oscillatory state-space models for temporal evolution in PINNs enable closed-form spatial differentiation and consistent boundary conditions while improving accuracy and cutting memory versus sequence models.

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

The paper introduces a physics-informed neural network that replaces general sequence models with linear-oscillator state-space dynamics to capture the time evolution of PDE solutions. This temporal component is paired with a PDE-aware spectral basis in space. The combination permits closed-form differentiation in space and straightforward enforcement of boundary conditions. Evaluations on forward, inverse, and high-dimensional PDE problems, including cases with up to 100 spatial dimensions, show gains in accuracy alongside lower memory use than recent sequence-model PINN baselines. The work therefore argues that structured dynamical priors aligned with PDE modal structure can make neural solvers both more accurate and more scalable.

Core claim

A PINN architecture that uses linear-oscillator-based state-space dynamics for temporal evolution together with a PDE-aware spectral basis in space achieves closed-form spatial differentiation, consistent boundary-condition enforcement, higher accuracy, and lower memory consumption than sequence-model-based PINN approaches when applied to forward, inverse, and high-dimensional time-dependent PDE problems up to 100 spatial dimensions.

What carries the argument

Linear-oscillator state-space model for temporal evolution combined with PDE-aware spectral basis for spatial representation, which together supply the structured inductive bias.

If this is right

  • Closed-form spatial differentiation becomes available without numerical approximation.
  • Boundary conditions can be enforced consistently across the domain.
  • Accuracy improves on both forward and inverse PDE problems relative to sequence-model baselines.
  • Memory requirements scale more favorably with sequence length and resolution.
  • The method remains applicable to problems with up to 100 spatial dimensions.

Where Pith is reading between the lines

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

  • The same oscillator prior could be tested on time-dependent systems outside the PDE setting, such as ODE networks or control problems.
  • Replacing the linear oscillator with a nonlinear state-space variant might extend the approach to problems with stronger nonlinear temporal dynamics.
  • Lower memory footprints could support longer-time or ensemble simulations that current sequence models cannot reach.
  • The spectral spatial basis might combine with other temporal priors, such as Hamiltonian or symplectic structures, to create further physics-aligned architectures.

Load-bearing premise

The temporal evolution of the target PDE solutions can be represented by linear-oscillator state-space dynamics without substantial loss of fidelity.

What would settle it

A time-dependent PDE whose solution exhibits strongly nonlinear or chaotic temporal behavior where the oscillatory state-space model produces lower accuracy or higher memory use than a comparable sequence-model PINN baseline.

Figures

Figures reproduced from arXiv: 2606.02623 by Abhishek Chandra, Taniya Kapoor.

Figure 1
Figure 1. Figure 1: Latent dynamics on convection (top) and wave (bottom). Columns: solution heatmap, [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: OSSM-PINN architecture: The initial condition is encoded into an oscilla￾tory LinOSS latent state, whose temporal roll￾out produces modal coefficients through an MLP decoder. Coefficients are combined with spatial basis and boundary factor to form the spatio-temporal solution, which is trained through physics-informed loss. This section introduces OSSM-PINNs for solv￾ing time-dependent PDEs. Let Ω ⊂ R d be… view at source ↗
Figure 3
Figure 3. Figure 3: Convection (β = 50): predicted u(x, t) fields and absolute errors for each method. 0.0 2.5 5.0 x 0.0 0.4 0.8 t Reference u(x, t) 0.8 0.0 0.8 0.0 0.4 0.8 t Predicted u(x, t) 0.0 2.5 5.0 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0.0 0.5 1.0 0.0 0.4 0.8 t Predicted u(x, t) 0.0 2.5 5.0 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0.0 0.5 1.0 0.0 0.4 0.8 t Predicted u(x, t) 0.0 2.5 5.0 x 0.0 0.4 0.8 t Absolute e… view at source ↗
Figure 4
Figure 4. Figure 4: Reaction: predicted u(x, t) fields and absolute errors for each method. benchmarks, with the largest gains on high-frequency convection, wave propagation, and Euler– Bernoulli beam problems. These results indicate that the oscillator–spectral factorization is effective across both first- and second-order-in-time PDEs, as well as high-order spatial operators. Figures 3 and 4, together with SM §E Figures 12,… view at source ↗
Figure 5
Figure 5. Figure 5: Frequency-domain comparison on the wave equation: predictions and Fourier spectra. 0 25 50 75 100 angular frequency ! 0.0 0.4 0.8 j^htopj conv ¯=50 conv ¯=100 wave EB 10 1 10 2 d o min a nt ! analytical ! OSSM-IM (FFT) OSSM-IMEX (FFT) analytical (bar) OSSM-IM (bar) OSSM-IMEX (bar) [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: SST inverse at t = 23 mo: reference (top), OSSM-IM prediction (mid), absolute error (bot). Full re￾sults in [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Schrödinger 100D: |ψ(x1, . . . , x100, t)| 2 at t = π/2 showing a slice in (x1, x2, x3). Reference (left), OSSM￾IM prediction (center), absolute error (right). L log Lip =37:9 PINNsFormer L log Lip =16:7 PINNMamba L log Lip =13:2 OSSM-PINN-IMEX [PITH_FULL_IMAGE:figures/full_fig_p009_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Forward-problem benchmark overview: key challenge and ground-truth solution for each [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Benchmark overview (inverse, geometry, high-dimensional, problem-adapted-basis). [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Wave equation: predicted u(x, t) fields and absolute errors for each method. 0 10 20 x 0.0 0.4 0.8 t Reference u(x, t) 0.8 0.0 0.8 0.0 0.4 0.8 t Predicted u(x, t) 0 10 20 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0 4 8 0.0 0.4 0.8 t Predicted u(x, t) 0 10 20 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0 4 8 0.0 0.4 0.8 t Predicted u(x, t) 0 10 20 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0 4 8 0.0 0.4 0.… view at source ↗
Figure 13
Figure 13. Figure 13: Euler–Bernoulli beam (extended domain [0, 8π]): predicted u(x, t) fields and absolute errors for each method. 0.0 2.5 5.0 x 0.0 0.4 0.8 t Reference u(x, t) 0.8 0.0 0.8 0.0 0.4 0.8 t Predicted u(x, t) 0.0 2.5 5.0 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0.0 0.4 0.8 0.0 0.4 0.8 t Predicted u(x, t) 0.0 2.5 5.0 x 0.0 0.4 0.8 t Absolute error 0.8 0.0 0.8 0.0 0.4 0.8 0.0 0.4 0.8 t Predicted u(x, t) 0.0 2.5 5.… view at source ↗
Figure 14
Figure 14. Figure 14: Convection (β = 100): predicted u(x, t) fields and absolute errors for each method. 29 [PITH_FULL_IMAGE:figures/full_fig_p029_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Euler–Bernoulli beam (classical, [0, 2π]): predicted u(x, t) fields and absolute errors for each method. 30 [PITH_FULL_IMAGE:figures/full_fig_p030_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: Latent dynamics on all benchmarks in addition to problems shown in Figure 1. Columns: [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Frequency-domain comparison on convection ( [PITH_FULL_IMAGE:figures/full_fig_p032_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Frequency-domain comparison on convection ( [PITH_FULL_IMAGE:figures/full_fig_p032_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Frequency-domain comparison on reaction at the final time: spatial Fourier magnitude [PITH_FULL_IMAGE:figures/full_fig_p032_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: Frequency-domain comparison on Euler–Bernoulli beam (classical, [PITH_FULL_IMAGE:figures/full_fig_p033_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Frequency-domain comparison on extended Euler–Bernoulli at the final time: spatial [PITH_FULL_IMAGE:figures/full_fig_p033_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: Loss-landscape on convection (β = 100) along top-2 Hessian eigenvectors. Smaller L log Lip indicates a smoother basin. 33 [PITH_FULL_IMAGE:figures/full_fig_p033_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: KdV inverse problem: predicted u(x, t) fields and absolute errors. OSSM-PINN recovers both coefficients (λ1, λ2) to < 0.1% relative error. Reference t = 6 mo t = 12 mo t = 18 mo t = 23 mo 1 0 1 T (°C) OSSM-IM: Pred 1 0 1 T (°C) OSSM-IM: |err| 0.00 0.01 0.02 |err| (°C) OSSM-IMEX: Pred 1 0 1 T (°C) OSSM-IMEX: |err| 0.00 0.01 0.02 |err| (°C) [PITH_FULL_IMAGE:figures/full_fig_p034_23.png] view at source ↗
Figure 24
Figure 24. Figure 24: SST 2D advection–diffusion inverse problem: predicted temperature [PITH_FULL_IMAGE:figures/full_fig_p034_24.png] view at source ↗
Figure 25
Figure 25. Figure 25: KdV coefficient convergence during training. Dashed line: true parameter value. [PITH_FULL_IMAGE:figures/full_fig_p034_25.png] view at source ↗
Figure 26
Figure 26. Figure 26: SST coefficient convergence during training ( [PITH_FULL_IMAGE:figures/full_fig_p035_26.png] view at source ↗
Figure 27
Figure 27. Figure 27: Frequency-domain validation of KdV inverse recovery at final time: spatial Fourier [PITH_FULL_IMAGE:figures/full_fig_p035_27.png] view at source ↗
Figure 28
Figure 28. Figure 28: Problem-adapted basis functions. 6 0 6 x 0.0 1.5 3.0 t Reference u(x, t) 0.6 0.0 0.6 0.0 1.5 3.0 t Predicted u(x, t) 6 0 6 x 0.0 1.5 3.0 t Absolute error 0.6 0.0 0.6 0.000 0.015 0.030 0.0 1.5 3.0 t Predicted u(x, t) 6 0 6 x 0.0 1.5 3.0 t Absolute error 0.6 0.0 0.6 0.000 0.015 0.030 0.0 1.5 3.0 t Predicted u(x, t) 6 0 6 x 0.0 1.5 3.0 t Absolute error 0.6 0.0 0.6 0.000 0.015 0.030 0.0 1.5 3.0 t Predicted u(… view at source ↗
Figure 29
Figure 29. Figure 29: QHO 1D: predicted fields with Fourier vs. Hermite basis. [PITH_FULL_IMAGE:figures/full_fig_p036_29.png] view at source ↗
Figure 30
Figure 30. Figure 30: Pöschl–Teller well: predicted fields with Fourier vs. PT basis. [PITH_FULL_IMAGE:figures/full_fig_p037_30.png] view at source ↗
Figure 31
Figure 31. Figure 31: Taylor–Green vortex 2D (Re = 100): predicted velocity components u(x, y), v(x, y) and pressure p(x, y), with absolute errors. 0.0 0.4 0.8 y Pred t=0.1 0.0 0.4 0.8 x 0.0 0.4 0.8 y |err| t=0.1 0.5 0.0 0.5 0.000 0.004 0.008 0.0 0.4 0.8 y Pred t=0.5 0.0 0.4 0.8 x 0.0 0.4 0.8 y |err| t=0.5 0.5 0.0 0.5 0.000 0.004 0.008 0.0 0.4 0.8 y Pred t=0.9 0.0 0.4 0.8 x 0.0 0.4 0.8 y |err| t=0.9 0.5 0.0 0.5 0.000 0.004 0.0… view at source ↗
Figure 32
Figure 32. Figure 32: Heat equation on a triangular domain: predicted [PITH_FULL_IMAGE:figures/full_fig_p037_32.png] view at source ↗
Figure 33
Figure 33. Figure 33: Schrödinger 5D: |ψ(x1, . . . , x5, t)| 2 at t = π/2, showing a slice in (x1, x2, x3). Reference, OSSM-IM prediction, and absolute error. Reference 0.616 0.624 0.632 OSSM-IM Pred 0.616 0.624 0.632 OSSM-IM |err| 0.0000 0.0002 0.0004 OSSM-IMEX Pred 0.616 0.624 0.632 OSSM-IMEX |err| 0.0000 0.0002 0.0004 Schrödinger 100D --- jÃj2 on three faces of the inner cube [¼=8; 7¼=8]3 ½ [0; ¼]3 at t = 1:57; remaining 97… view at source ↗
Figure 34
Figure 34. Figure 34: Schrödinger 100D: |ψ(x1, . . . , x100, t)| 2 at t = π/2, showing a slice in (x1, x2, x3). Reference, OSSM-IM prediction, and absolute error. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_34.png] view at source ↗
Figure 35
Figure 35. Figure 35: Training-loss histories for OSSM-PINN-IMEX (three seeds). [PITH_FULL_IMAGE:figures/full_fig_p039_35.png] view at source ↗
Figure 36
Figure 36. Figure 36: Training-loss histories for OSSM-PINN-IM (three seeds). [PITH_FULL_IMAGE:figures/full_fig_p040_36.png] view at source ↗
Figure 37
Figure 37. Figure 37: Seed stability: mean rMAE ±1 std over three seeds for seven benchmarks. 40 [PITH_FULL_IMAGE:figures/full_fig_p040_37.png] view at source ↗
read the original abstract

Solving time-dependent partial differential equations (PDEs) is an important problem in computational science and engineering. Physics-informed neural networks (PINNs) learn PDE solutions from governing equations. However, accurately capturing temporal evolution remains challenging. Recent sequence-model-based approaches parameterize time evolution using general-purpose sequence models, which capture temporal dependencies but do not explicitly encode the structured dynamics of PDE solutions. In addition, their memory requirements can scale unfavorably with sequence length and resolution, limiting applicability in large-scale or high-dimensional settings. This work introduces a PINN approach that incorporates oscillatory state-space dynamics to represent the modal structure of PDE solutions. The proposed method leverages a linear-oscillator-based temporal evolution, together with a PDE-aware spectral basis in space. This design enables closed-form spatial differentiation and consistent enforcement of boundary conditions. The method is evaluated on forward, inverse, and high-dimensional PDE problems, including cases up to 100 spatial dimensions. The results show improved accuracy and reduced memory usage compared to recent sequence-model-based PINN approaches. Overall, this work highlights the benefits of incorporating structured dynamical priors into the temporal evolution of neural PDE solvers and suggests designing more physics-aligned and computationally efficient PINN architectures.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript proposes a physics-informed neural network (PINN) architecture that uses oscillatory state-space models to parameterize temporal evolution of PDE solutions, paired with a PDE-aware spectral basis for spatial discretization. This enables closed-form spatial derivatives and boundary condition enforcement. The approach is evaluated on forward, inverse, and high-dimensional (up to 100D) PDE problems and is reported to outperform recent sequence-model-based PINNs in accuracy while using less memory.

Significance. If the empirical gains hold under rigorous verification, the work would demonstrate the value of embedding structured linear dynamical priors into PINN temporal modules, offering a route to scalable solvers for high-dimensional time-dependent PDEs where general sequence models become memory-intensive.

major comments (2)
  1. [Abstract] Abstract: The central claim that linear-oscillator state-space dynamics represent the modal temporal evolution 'without significant loss of fidelity' for the target PDEs is load-bearing, yet the abstract provides no indication of how the model is modified or regularized when the underlying PDE exhibits damping, exponential decay, or strong nonlinearity (e.g., parabolic or chaotic regimes).
  2. [Abstract] Abstract: The reported accuracy and memory improvements are presented without reference to specific PDE families tested, sequence lengths, spatial resolutions, or quantitative baselines (error bars, number of runs), making it impossible to assess whether the gains are robust or confined to pre-selected oscillatory problems.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed comments on the abstract. We respond point by point below and will revise the abstract for clarity where appropriate.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The central claim that linear-oscillator state-space dynamics represent the modal temporal evolution 'without significant loss of fidelity' for the target PDEs is load-bearing, yet the abstract provides no indication of how the model is modified or regularized when the underlying PDE exhibits damping, exponential decay, or strong nonlinearity (e.g., parabolic or chaotic regimes).

    Authors: The manuscript positions the oscillatory state-space model as an inductive bias specifically for PDEs whose solutions exhibit modal oscillatory structure (see Introduction and Section 3). No explicit modification or regularization for damping/strong nonlinearity is introduced because the target problems are those where the linear oscillator prior aligns with the physics; applicability outside this regime is discussed as a limitation in the conclusion. We agree the abstract should better delimit scope and will revise it to state that the approach targets oscillatory modal evolution. revision: yes

  2. Referee: [Abstract] Abstract: The reported accuracy and memory improvements are presented without reference to specific PDE families tested, sequence lengths, spatial resolutions, or quantitative baselines (error bars, number of runs), making it impossible to assess whether the gains are robust or confined to pre-selected oscillatory problems.

    Authors: The abstract is intentionally high-level; concrete PDE families (wave, Schrödinger, etc.), sequence lengths, resolutions, and quantitative results (including error bars over multiple runs) appear in Section 4 and the associated tables/figures. We acknowledge that the abstract could better signal the breadth of evaluation and will revise it to name example PDE families and note that results include statistical quantification over repeated trials. revision: yes

Circularity Check

0 steps flagged

No circularity: empirical method proposal with independent evaluation

full rationale

The paper introduces an oscillatory state-space model as an inductive bias for PINNs, combined with a spectral spatial basis. The abstract and description frame this as an architectural choice evaluated empirically on forward/inverse/high-dimensional PDE tasks, with reported gains in accuracy and memory. No equations, fitting procedures, or derivation steps are presented that reduce a claimed prediction or result to a fitted input, self-definition, or self-citation chain. The central claim remains an empirical improvement over sequence-model baselines rather than a self-referential derivation. This is a standard self-contained proposal; no load-bearing step collapses by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract supplies no explicit free parameters, axioms, or invented entities; the central claim rests on the unstated premise that linear oscillators capture the modal structure of the target PDEs.

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discussion (0)

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