Sub-Sequential Physics-Informed Learning with State Space Model

In the past few years, Physics-Informed Neural Networks (PINNs) (Raissi et al., 2019) have emerged as a novel approach for numerically solving partial differential equations (PDEs). PINN takes a neural network $u_{\theta}(x,t)$, whose parameters $\theta$ are trained with physics PDE residual loss, as the numerical solution $u(x,t)$ of the PDE, where $x$ and $t$ are spatial and temporal coordinates. The core idea behind PINNs is to take advantage of the universal approximation property of neural networks (Hornik et al., 1989) and automatic differentiation implemented by mainstream deep learning frameworks, such as PyTorch (Paszke et al., 2019) and Tensorflow (Abadi et al., 2016), so that PINNs can achieve potentially more precise and efficient PDE solution approximation compared with traditional numerical approaches like finite element methods (Reddy, 1993).

The mainstream PINNs predominantly employ multilayer perceptrons (MLPs) as their backbone architecture. However, despite the universal approximation capability of MLPs, they do not always guarantee the accurate learning of numerical solutions to PDEs in practice. This phenomenon is observed as the failure modes in PINNs, in which case PINN provides a completely wrong approximation (Krishnapriyan et al., 2021). As illustrated in Fig. 1, the failure modes often manifest as a temporal gradual distortion. This distortion arises because MLPs lack the necessary inductive bias to effectively capture the temporal dependencies of a system, ultimately hindering the accurate propagation of physical patterns informed by the initial conditions.

To introduce such an inductive bias, several sequence-to-sequence approaches have been proposed (Krishnapriyan et al., 2021; Zhao et al., 2024; Yang et al., 2022; Gao et al., 2022). Specifically, Krishnapriyan et al. (2021) propose training a new network at each time step and recursively using its output as the initial condition for the next step. This method incurs significant computational and memory overhead while exhibiting poor generalization. Furthermore, Transformer-based approaches (Zhao et al., 2024; Yang et al., 2022; Gao et al., 2022) are proposed to address the time-dependency issue. Yet, these methods are based on discrete sequences, making their model produce incorrect pattern mutations in some cases due to their ignorance of the basic principle that PINNs approximate continuous dynamical systems. Thus, there is still an open question:

How can we effectively introduce sequentiality to PINNs?

To answer this question, we need to understand the essential difficulties of training PINNs. First, PINNs assume temporal continuity, whereas, during their actual training, the spatio-temporal collection points used to construct the PDE residual loss are sampled discretely. We define this nature of PINN as Continuous-Discrete Mismatch. In the absence of a well-defined continuous-discrete articulation, the real trajectory of physical system is not necessarily recovered correctly in the training process, since such Continuous-Discrete Mismatch would block the propagation of the initial condition.

To respect the inherent Continuous-Discrete Mismatch, we reveal that the State Space Models (SSM) (Kalman, 1960) can be a good continuous-discrete articulation. SSMs parametrically model a discrete temporal sequence as a continuous dynamic system. An SSM’s discrete form approximates the instantaneous states and rates of change of its continuous form via integrating the system’s dynamics over each time interval, which more accurately responds to the trajectory of a continuous system. Meanwhile, the SSM unifies the scale of derivatives of different orders, making it easier to be optimized. So far, SSMs have shown their insane capacity in language (Gu & Dao, 2023) and vision (Liu et al., 2024a) tasks, but its potential for solving PDEs remains unexplored. We propose to construct PINNs with SSMs to unleash their excellent properties of continuous-discrete articulation.

Next, we remark that the simplicity bias (Shah et al., 2020) of neural networks is another crucial contributing factor to PINN training difficulty. The simplicity bias will lead the model to choose the pattern with the simplest hypothesis. This results in an over-smoothed solution when approximating PDEs. Because, for data-free PINNs, there might be a very simple function in the feasible domain that can make the residual loss zero. For example, for convection equation, $\bar{u}(x,t)=0$ leads to zero empirical loss on every collection point except when $t=0$. While the correct pattern is hard to fight against over-smoothed patterns during training.

A major way to eliminate simplicity bias is to construct agreements over diversity predictions (Teney et al., 2022). Following this principle, we propose a novel sub-sequence alignment approach, which allows the diverse predictions of time-varying SSMs to form such agreements. Sub-sequence modeling adopts a medium sequence granularity, forming the time dependency that a small sequence fails to capture while avoiding the optimization problem along with the long sequence. Meanwhile, the alignment of the sub-sequence predictions ensures the global pattern propagation and the formation of an agreement that eliminates simplicity bias.

In this paper, we introduce a novel learning framework to solve physics PDE’s numerically, named PINNMamba, which performs time sub-sequences modeling with the Selective SSMs (Mamba) (Gu & Dao, 2023). PINNMamba successfully captures the temporal dependence within the PDE when training the continuous dynamical systems with discretized collection points. To the best of our knowledge, PINNMamba is the first data-free SSM-based model that effectively solve physics PDE. Experiments show that PINNMamba outperforms other PINN approaches such as PINNsFormer (Zhao et al., 2024) and KAN (Liu et al., 2024c) on multiple hard problems, achieving a new state-of-the-art.

Contributions. We make the following contributions:

• •

  We reveal that the mismatch between the discrete nature of the training collection points and the continuous nature of the function approximated by the PINNs is an important factor that prevents the propagation of the initial condition pattern over time in PINNs.

• •

  We also note that the simplicity bias of neural networks is a key contributing factor to the over-smoothing pattern that causes gradual distortion in PINNs.

• •

  We propose PINNMamba, which eliminates the discrete-continuity mismatch with SSM and combats simplicity bias with sub-sequential modeling, resulting in state-of-the-art on several PINN benchmarks.

Due to page limit, we discuss related works in Appendix A.

Physics-Informed Neural Networks. The PDE systems that are defined on spatio-temporal set $\Omega\times[0,T]\subseteq\mathbb{R}^{d+1}$ and described by equation constraints, boundary conditions, and initial conditions can be formulated as:

where $u:\mathbb{R}^{d+1}\rightarrow\mathbb{R}^{m}$ is the solution of the PDE, $x\in\Omega$ is the spatial coordinate, $\partial\Omega$ is the boundary of $\Omega$, $t\in[0,T]$ is the temporal coordinate and $T$ is the time horizon. The $\mathcal{F},\mathcal{I},\mathcal{B}$ denote the operators defined by PDE equations, initial conditions, and boundary conditions respectively.

A physics-driven PINN first builds a finite collection point set $\chi\subset\Omega\times[0,T]$, and its spatio (temporal) boundary $\partial\chi\subset\partial\Omega\times[0,T]$ ($\chi_{0}\subset\Omega\times\{0\}$), then employs a neural network $u_{\theta}(x,t)$ which is parameterized by $\theta$ to approximate $u(x,t)$ by optimizing the residual loss as defined in Eq. 7:

where $\lambda_{\mathcal{F}}$,$\lambda_{\mathcal{I}}$,$\lambda_{\mathcal{B}}$ are the weights for loss that are adjustable by auto-balancing or hyperparameters. $\|\cdot\|$ denotes $l^{2}$-norm.

State Space Models. An SSM describes and analyzes a continuous dynamic system. It is typically described by:

where $\mathbf{h}(t)$ is hidden state of time $t$, $\mathbf{\dot{h}}(t)$ is the derivative of $\mathbf{h}(t)$. $\mathbf{x}(t)$ is the input state of time $t$, $\mathbf{u}(t)$ is the output state, and $A,B,C,D$ are state transition matrices.

In real-world applications, we can only sample in discrete time for building a deep SSM model. We usually omit the term $D\mathbf{x}(t)$ in deep SSM models because it can be easily implemented by residual connection (He et al., 2016). So we create a discrete time counterpart:

with discretization rules such as zero-order hold (ZOH):

where $\bar{A}$ and $\bar{B}$ is discrete time state transfer and input matrix, and $\mathrm{\Delta}$ is a step size parameter. By parameterizing $A$ using HiPPO matrix (Gu et al., 2020), and parameterizing $(\Delta,B,C)$ with input-dependency, a time-varying Selective SSM can be constructed (Gu & Dao, 2023). Such a Selective SSM can capture the long-time continual dependencies in dynamic systems. We will argue that this makes SSM a good continuous-discrete articulation for modeling PINN.

A counterintuitive fact of PINNs is that the failure modes are not devoid of optimizing their residual loss to a very low degree. As shown in Fig. 2, for the convection equation, the converged PINN almost completely crashes in the domain, but its loss maintains a near-zero degree at almost every collection point. This is the result of the combined effects of the simplicity bias (Shah et al., 2020; Pezeshki et al., 2021) of neural networks and the Continuous-Discrete Mismatch of PINNs, as shown in Fig. 3. The simplicity bias is the phenomenon that the model tends to choose the one with the simplest hypothesis among all feasible solutions, which we demonstrate in Fig. 3 (b). Continuous-Discrete Mismatch refers to the inconsistency between the continuity of the PDE and the discretization of PINN’s training process. As shown in Eq. 4 - 6, to construct the empirical loss for the PINN training process, we need to determine a discrete and finite set of collection points on $\Omega\times[0,T]$. This is usually done with a grid or uniform sampling. But a PDE system is usually continuous and its solutions should be regular enough to satisfy the differential operator $\mathcal{F}$, $\mathcal{B}$, and $\mathcal{I}$.

Continuous-Discrete Mismatch. Continuous-Discrete Mismatch will cause correct local patterns hard to propagate over the global domain. Because the loss on discrete collection points does not necessarily respond to the correct pattern on the continuous domain, instead, only responds to its small neighborhood. To show such Continuous-Discrete Mismatch, we first present the following theorem:

By Theorem 3.1, enforcing the PDE only at a finite set of points does not guarantee a globally correct solution. This can be performed by simply constructing a Bump function in a small neighborhood of points in $\chi^{*}$ so that it satisfies $\mathcal{M}(u_{\theta}(x^{*},t^{*}))=0$ for $(x^{*},t^{*})\in\chi^{*}$. This means that the information of the equation determined by the initial conditional differential operator $\mathcal{I}$ may act only on a small neighborhood of collection points with $t=0$. The other collection points in the $\Omega\times(0,T]$, on the other hand, might fall into a local optimum that can make $\mathcal{L}_{\mathcal{F}}(u_{\theta})$ defined by Eq. 4 to near 0. Because the function $u_{\theta}$ determined by $\mathcal{F}$ and $\mathcal{I}$ together on the collection points at $t=0$ may not be generalized outside its small neighborhood. The detailed proof of Theorem 3.1 can be found in Appendix B.

Simplicity Bias. Meanwhile, the simplicity bias of neural networks will make the PINNs always tend to choose the simplest solution in optimizing $\mathcal{L}_{\mathcal{F}}(u_{\theta})$. This implies that PINN will easily fall into an over-smoothed solution. For example, as shown in Fig. 2, the PINN’s prediction is 0 in most regions. The loss of this over-smoothed feasible solution is almost identical to that of the true solution, and the existence of an insurmountable ridge between the two loss basins results in a PINN that is extremely susceptible to falling into local optimums. As in Fig 3, the over-smoothed pattern yields an advantage against the correct pattern.

Under the effect of difficulty in passing locally correct patterns to the global due to Continuous-Discrete Mismatch and over-smoothing due to simplicity bias, PINNs present failure modes. Therefore, to address such failure modes, the key points in designing the PINN models lie in: (1) a mechanism for information propagation in continuous time and (2) a mechanism to eliminate the simplicity bias of models.

To address the problems in Section 3, we propose (1) a discrete state-space-based encoder that models the sequences of individual collection points in continuous dynamics, to match with Continuous-Discrete Mismatch, and propagates the information from the initial condition to subsequent times (Section 4.1). and (2) a sub-sequence contrastive alignment mechanism that aligns different outputs of the same collection point in different sub-sequences, to form an agreement that eliminates simplicity bias (Section 4.2).

In conjunction with the high-level ideas described in Section 4, in this section, we present PINNMamba, a novel physics-informed learning framework that effectively combats the failure modes in the PINNs.

Sub-Sequential I/O. As shown in Fig. 4, PINNMamba first samples the grid of collection points over the entire spatio-temporal domain bounded by the PDE. We assume that the grid picks $M$ spatial coordinates and $N$ temporal coordinates, and denote the temporal sampling interval as $\Delta t=T/(N-1)$. For a collection point $(x,t)$, we construct a sequence $X(x,t)$ with its $k-1$ temporal successors:

PINNMamba takes such $M\times N$ sequences as the input of models. For each sequence $X(x,t)$, PINNMamba computes a sub-sequence prediction $\{\bar{u}_{\theta}^{t}(x,t),\bar{u}_{\theta}^{t}(x,t+\Delta t),\cdots,\bar{u}_{\theta}^{t}(x,t+(k-1)\Delta t)\}$ corresponding to every collection point in the sequence, where $\bar{u}_{\theta}^{t}(x,t+i\Delta t)$ denote the tentative prediction of collection point $(x,t+i\Delta t)$ in a sequence start with time $t$. The $\bar{u}_{\theta}^{t}(x,t)$ will be taken as the output of collection point $(x,t)$ and the rest of the sequence will be used to construct the sub-sequence contrastive alignment loss we will discuss later in Section 4.2. The residual losses of the model w.r.t the sub-sequence will be:

Model Architecture. As shown in Fig. 4, PINNMamba employs an encoder-only architecture, which encodes fixed-size input sub-sequence into a sub-sequence prediction with the same length. First, for each token in the sequence, an MLP-based Spatio-Temporal Embedding layer first embeds the $(x,t)$ coordinates into high-dimensional representation. The embeddings will be sent to a Mamba-based encoder, which consists of several PINNMamba blocks.

The PINNMamba block employed here consists of two branches: (1) the first is a stack of a linear projection layer, a 1d-convolution layer, a Wavelet activation (Zhao et al., 2024), and an SSM layer with parallel scan (Gu & Dao, 2023); (2) the second is a stake of a linear projection layer and a Wavelet activation. The two branches are then connected with an element-wise multiplication, followed by another linear projection and residual connection. With input $X^{l}$, the PINNMamba block can be formulated as:

where $\sigma(x)=\omega_{1}\sin(x)+\omega_{2}\cos(x)$ is Wavelet activation function (Zhao et al., 2024), in which $\omega_{1},\omega_{2}$ are learnable. $\otimes$ denotes an element-wise multiplication.

Sub-Sequence Contrastive Alignment. PINNMamba predicts the same collection multiple times in different subsequences. For example, the collection point $(x,k+\Delta t)$ appears on sequences from $X(x,t+\Delta t)$ to $X(x,t+k\Delta t)$. We align the predictions on these subsequences to make the information defined by the initial conditions propagate over time. To do this, for each subsequence, we design a contrastive loss with the last subsequence for alignment:

Thus, the empirical loss for PINNMamba is defined as:

Setup. We evaluate the performance of PINNMamba on three standard PDE benchmarks: convection, wave, and reaction equations, all of which are identified as being affected by failure modes (Krishnapriyan et al., 2021; Zhao et al., 2024). The details of those PDEs can be found in Appendix D. We compare PINNMamba with four baseline models, vanilla PINN (Raissi et al., 2019), QRes (Bu & Karpatne, 2021), PINNsFormer (Zhao et al., 2024), and KAN (Liu et al., 2024c) . For fair comparison, we sample 101$\times$101 collection points with uniformly grid sampling, following previous work (Zhao et al., 2024; Wu et al., 2024). We also evaluate on PINNacle Benchmark (Hao et al., 2024) and Navier–Stokes equation (Raissi et al., 2019).

Training Details. We train PINNMamba and all the baseline models 1000 epochs with L-BFGS optimizer (Liu & Nocedal, 1989). We set the sub-sequence length to 7 for PINNMamba, and keep the original pseudo-sequence setup for PINNsFormers. The weights of loss terms $[\lambda_{\mathcal{F}},\lambda_{\mathcal{I}},\lambda_{\mathcal{B}}]$ are set to $[1,1,10]$ for all three equations, as we find that strengthening the boundary conditions can lead to better convergence. $\lambda_{\text{alig}}$ is set to 1000 for convection and reaction equations, and auto-adapted by $\lambda_{\mathcal{F}}$ for wave equation. All experiments are implemented in PyTorch 2.1.1 and trained on an NVIDIA H100 GPU. More training details are in Appendix E. Our code and weights are available at \urlhttps://github.com/miniHuiHui/PINNMamba.

Metrics. To evaluate the performance of the models, we take relative Mean Absolute Error (rMAE, a.k.a $\ell_{1}$ relative error) and relative Root Mean Square Error (rRMSE, a.k.a $\ell_{2}$ relative error) following common practive (Zhao et al., 2024; Wu et al., 2024). The metrics are formulated as:

where N is the number of test points, $u(x,t)$ is the ground truth solution, and $\hat{u}(x,t)$ is the model’s prediction.

In this paper, we reveal that the mismatch between discrete training of PINNs and the continuous nature of PDEs, as well as simplicity bias are the key of failure modes. In combating with such failure modes, we propose PINNMamba, an SSM-based sub-sequence learning framework. PINNMamba successfully eliminates the failure modes, and meanwhile becomes the new state-of-the-art PINN architecture.

The development of physics-informed neural networks represents a transformative approach to solving differential equations by integrating physical laws directly into the learning process. This work explores novel advancements in PINN architecture, to improve accuracy and eliminate the potential failure modes. By refining PINN architectures, this study contributes to the broader adoption of physics-informed machine learning in fields such as computational fluid dynamics, material science, and engineering simulations. The proposed enhancements lead to more robust and scalable models, facilitating real-world applications where conventional PINNs struggle with over-smoothing. There is no known negative impact from this study at this time.