De Rham cohomological architectures for PINNs and PDEs with topological holes

Andrew Gracyk

cs.LGNov 3, 2026

De Rham cohomological architectures for PINNs and PDEs with topological holes

Authors: Andrew Gracyk

Organizations: Department of Mathematics, Purdue University, West Lafayette, IN 47907, United States

Abstract

We attempt to alleviate failure modes of classical physics-informed neural networks (PINNs) in scenarios of topological pathologies corresponding to punctures, holes, poles, and voids by introducing de Rham cohomological augmentations, or Hodge-de Rham PINNs. Vanilla PINNs often encounter pitfalls in hole-based settings due to discontinuous interfaces, nonzero closed path integration, path dependence, correspondence to exact forms via the exterior derivative, etc., therefore we absolve traditional failures via parameterizing a 1-form by the Hodge-Morrey-Friedrichs (HMF) decomposition for manifolds with boundary, drawing a connection to de Rham cohomology such as through the HMF isomorphism theorem. 1-forms can act as representations of scalar-valued PDE solutions through integral operator mappings. We propose performing a PINN-based loss on the parameterized 1-form through an associated PDE, which subsequently solves the potential-based PINN problem and moreover remedies the failures of vanilla PINNs via the Sobolev-regular operator under a Fréchet differentiation interconnection. The operator relationship often has a nontrivial closed-form and often requires integration or a case-by-case analysis, which we address thoroughly in our experiments. When an integrability condition is not met on our losses, we can equip exhaustion functions to train the networks. We demonstrate our methods on the Aharonov-Bohm effect, which, to our understanding, is largely nonexistent in PINN literature. We develop theoretical guarantees of our methods and demonstrate our pathological safeguards on a broad selection of topologically-adverse problems, including scenarios of inexact forms, and those of nontrivial harmonics and coexact terms.

Figures & tables

Figure 1: We plot success of a (1) Hodge-de Rham PINN and failure of a (2) vanilla PINN on a domain $\{(x,y)|1<x^{2}+y^{2}<4\}$ with a topological hole under the PDE $\displaystyle\Delta u=\frac{\partial^{2}u}{\partial x^{2}}+\frac{\partial^{2}u}{\partial y^{2}}=0,$ where $\Delta$ is the Laplacian with respect to the flat metric. The true solution is (3). We use the loss $\mathcal{L}=\|\Delta u\|_{L^{2}(\Omega^{*};d\mu_{\Omega}^{*})}^{2}+\left\|\frac{\partial u}{\partial x}x+\frac{\partial u}{\partial y}y\right\|_{L^{2}(\Gamma_{\text{in}}^{*};d\mu_{\text{in}}^{*})}^{2}+\left\|\nabla u-v_{\text{target}}\right\|_{L^{2}(\Gamma_{\text{out}}^{*};d\mu_{\text{out}}^{*})}^{2}.$ We train for 12,000 epochs and use a Fourier feature embedding as in Wang et al. (2023) with $\sigma=1.0$ . We see complete failure of the PINN. By the Fundamental Theorem of Line Integrals, we have $\displaystyle\oint_{\Gamma_{\text{out}}}\nabla u\cdot dl=u(\text{start})-u(\text{end})=0,$ but $v_{\text{target}}$ corresponds to a non-exact form. This PINN completely fails (as opposed to only failing at the discontinuity) because Neumann boundary conditions are used.

Figure 2: We illustrate our architecture as in 3.2 . We specialize to the case of $(x,y)\in\mathbb{R}^{2}$ .

Figure 3: We plot (1) success of a Hodge-de Rham PINN versus (2) the true scalar solution with (3) pointwise absolute error after about $\sim 30,000$ epochs on the Aharonov-Bohm experiment. We found a vanilla MLP worked better without a Fourier feature embedding. We lower the learning rate in late training.

Figure 4: We plot (1) catastrophic failure of a vanilla PINN versus (2) the true scalar solution with (3) pointwise absolute error after about $\sim 12,000$ epochs on the Aharonov-Bohm experiment.

Figure 5: We plot (1) success of a Hodge-de Rham PINN versus (2) the true scalar solution with (3) pointwise absolute error after about $\sim 30,000$ epochs on the Aharonov-Bohm experiment with two poles. We lower the learning rate in late training.

Figure 6: We plot (1) catastrophic failure of a vanilla PINN versus (2) the true scalar solution with (3) pointwise absolute error after about $\sim 12,000$ epochs on the Aharonov-Bohm experiment with two poles.

Figure 7: We plot (1) catastrophic failure of a canonically-developed vector-valued PINN (so the neural network is now vector-valued) versus (2) the true scalar solution with (3) pointwise absolute error after about $\sim 12,000$ epochs on the Aharonov-Bohm experiment with two poles.

Symbol	Description
$\alpha$	1-form (smooth section of the cotangent bundle)
$\Omega$	Domain/manifold without puncture
$\Omega^{*}$	Domain/manifold with puncture(s)
$\Phi$	Differential operator of the scalar PDE
$\Psi$	Differential operator of the PDE on the 1-form
$[\alpha]$	Cohomology class containing $\alpha$

Table 8

Appendix figures & tables25 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 8: We plot that Theorem 3 holds on the unit radial vortex experiment with simple boundary condition. We plot a log-log plot demonstrating the bound of $\left|\oint_{\gamma_{i}}\alpha_{\theta}-\oint_{\gamma_{i}}\alpha^{\star}\right|\leq\widetilde{C}_{i}\|\alpha_{\theta}-\alpha^{\star}\|_{H^{s}(M;\Lambda^{1})}$ holds. We plot the errors across three distinct closed curves $\gamma_{i}$ , which are concentric circles with radii $r\in\{1.1,1.5,1.9\}$ with respect to 8,000 training epochs. The x-axis is the $H^{s}$ form error and the y-axis is the path circulation error. The shaded region represents the desirable region such that the bound holds. The points are clustered in the low-error regime since this is where the training spends most of its time (loss converges quickly then slowly).

Figure 9: We plot the coefficient $c(t)$ after 3,000 epochs at 5 discrete $t$ values for the arctangent without zero experiment with (left) a typical Hodge-decomposed architecture; and (right) a specialized Hodge-decomposed architecture where a $e^{x}$ deformation is baked into architecture, so the ansatz is $A=e^{-x}\alpha_{\theta}=e^{-x}\nabla\phi+e^{-x}\nabla\times\beta+c(t)h$ and $\alpha_{\theta}$ is a Hodge-decomposed learned form. As we can see, the true values of $c$ are learned much more precisely with the architecture specialized for this experiment.

Figure 10: We illustrate (1) success with 1-form learning; (2) the true solution; and (3) pointwise absolute error on the Kähler manifold experiment with a Pythagorean loss function.

Figure 11: We plot the value of $c$ (there is one puncture, so one value) corresponding to our unit radius vortex experiment. We plot (green) the learned value where the learning rate of the $c$ parameter is increased to $5\text{e}{-2}$ and the remaining networks use learning rate $1\text{e}{-3}$ , whereas the learning rate for (red) is the same as the remaining networks with $1\text{e}{-3}$ . We can see the high learning rate case approaches a value of $2\pi$ , which is the anticipated true learned value in this experiment.

Figure 12: We plot the training losses over 3,000 epochs on our unit radial vortex experiment with (1) a relatively low learning rate of $1\text{e}{-3}$ for all $\phi_{\theta},\beta_{\theta},c_{\theta}$ ; and (2) a learning rate of $1\text{e}{-3}$ for $\phi_{\theta},\beta_{\theta}$ and a high learning rate of $5\text{e}{-2}$ for $c_{\theta}$ only. We set loss coefficients $\lambda_{\text{physics}}=1,\lambda_{\text{boundary}}=1$ .

Figure 13: We plot training loss on 3,000 epochs on the arctangent without zero experiment with a vanilla Hodge-decomposed ansatz $\alpha_{\theta}$ and a specialized ansatz $e^{-x}\alpha_{\theta}$ tailored for this specific problem. The tailored ansatz clearly outperforms. This figure complements 9 .

Figure 14: We illustrate (1) success with 1-form learning, (2) catastrophic failure of a vanilla PINN, and (3) the true solution on the unit radial vortex experiment with complicated boundary condition.

Figure 15: We illustrate (1) catastrophic failure of a canonically-developed vector-valued PINN; (2) the true solution; (3) pointwise absolute error after about $\sim 12,000$ epochs.

Figure 16: We plot failure of a vanilla PINN on the arctangent without zero experiment. We plot (1) the PINN solution; (2) the true solution; (3) the absolute pointwise error. We train the PINN for $\sim 8,000$ epochs, and the loss has plateaued. As we can see, the discontinuity is not learned, compatible with the discussion in 3.1 .

Figure 17: We plot success of the Hodge-de Rham PINN on the arctangent without zero experiment. We plot (1) the PINN solution; (2) the true solution; (3) the absolute pointwise error. We train the PINN for $3,000$ epochs.

Figure 18: We plot the vector field according to the 1-form $\alpha=\alpha_{1}dx+\alpha_{2}dy$ on the arctangent without zero experiment.

Figure 19: We plot training loss on the arctangent without zero experiment when we set the radius around the pole to $r=0$ and we exclude an exhaustion function. As we can see, the loss is highly stochastic and blows-up.

Figure 20: We plot training loss on the arctangent with exhaustion experiment when we set the radius around the pole to $r=0$ and we include an exhaustion function. As we can see, the loss is stable.

Figure 21: We plot partial success, partial failure of the arctangent without zero (exhaustion function excluded) experiment after 4,000 epochs but we unrestrict $\{(x,y)|x^{2}+r^{2}<0.2\}$ on the test dataset, but it is omitted in the training data. We plot (1) the PINN solution; (2) the true solution; (3) absolute pointwise error. The learning on the previously-omitted annular region is solely due to out-of-distribution learning.

Figure 22: We plot the same figure as 21 but with exhaustion after 4,000 epochs. We plot (1) the Hodge-de Rham PINN solution; (2) the true solution; (3) absolute pointwise error. As we can see, there is significantly greater success since maximum pointwise error drops from $\sim 0.7\rightarrow\sim 0.15$ .

Figure 23: We illustrate (1) catastrophic failure of a vanilla PINN; (2) the true solution; and (3) pointwise absolute error on the Kähler manifold experiment as in Appendix H.1 with a Pythagorean loss function after $\sim 7,000$ epochs.

Figure 24: We illustrate (1) success of a vector-valued vanilla PINN; (2) the true solution; and (3) pointwise absolute error on the Kähler manifold experiment as in Appendix H.1 with a Pythagorean loss function after $\sim 12,000$ epochs.

Figure 25: We sample circles and compute closed line integrals as in the Kähler (complex manifold domain) experiment as in Appendix H.1 on the true solution. We sample circles around a pole and compute the circulation value. The left demonstrates the sampled circles around the pole, and the right demonstrates the corresponding values. Color is used to demonstrate radii values. As we can see on the right, the circulation is nonzero and approaches the theoretical value $\Gamma_{1}$ .

Figure 26: We plot the same as Figure 25 but we perform the sampling on the scalar-valued PINN solution that fails and not the true solution. As we can see, the corresponding circulation is zero.

Figure 27: We illustrate (1) catastrophic failure of a vector-valued PINN; (2) the true solution; and (3) pointwise absolute error on the Kähler manifold experiment as in Appendix H.2 with a Pythagorean loss function after $\sim 8,000$ epochs.

Figure 28: We illustrate (1) success of a Hodge-de Rham PINN; (2) the true solution; and (3) pointwise absolute error on the Kähler manifold experiment as in Appendix H.2 with a Pythagorean loss function after $\sim 2,000$ epochs.

Figure 29: We plot the difference in value between two paths from one point to another, possibly encompassing poles. We plot (1) a toy diagram illustrating what we convey; (2) the difference in path values for a vector-valued vanilla PINN; (3) the difference in path values for the true solution. This figure illustrates the vector-valued PINN solution exhibits different path-dependent behavior than the true solution.

Figure 30: We illustrate that the Aharonov-Bohm experiment true solution corresponds to a non-exact form. In particular, the bottom left panel would be exactly zero if the exterior derivative corresponded to an exact form, which is what the bottom right panel demonstrates.

Figure 31: We plot the Aharonov-Bohm experiment with two poles image across training iteration in (top) 200; (middle) 500; (bottom) 1,000 epoch chunks. The branch cut at initialization is due to the harmonic term. In particular, the branch cut at initialization on the left-hand side is intentional to facilitate the learning of the branch cut after training. Note that we use $\text{sin}(\cdot)$ activation. We do not use a Fourier feature embedding as in Tancik et al. (2020) Wang et al. (2023)

Figure 32: We provide a supplemental experiment of success of our 1-form methods on the Kähler manifold experiment with six punctures to illustrate success of our methods on scenarios of many punctures.

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