PINP: Physics-Informed Neural Predictor with latent estimation of fluid flows
Authors: Huaguan Chen
Yang Liu
Hao Sun
Venue: ICLR 2025 Poster (2025)
Keywords: Fluid dynamics
Spatiotemporal prediction
Physics-informed learning
Primary area: applications to physical sciences (physics, chemistry, biology, etc.)
TL;DR: We integrated PDEs into the network and loss function, enabling the prediction of observable physical quantities and the inference of future latent physical quantities as interpretation.

Abstract: Accurately predicting fluid dynamics and evolution has been a long-standing challenge in physical sciences. Conventional deep learning methods often rely on the nonlinear modeling capabilities of neural networks to establish mappings between past and future states, overlooking the fluid dynamics, or only modeling the velocity field, neglecting the coupling of multiple physical quantities. In this paper, we propose a new physics-informed learning approach that incorporates coupled physical quantities into the prediction process to assist with forecasting. Central to our method lies in the discretization of physical equations, which are directly integrated into the model architecture and loss function. This integration enables the model to provide robust, long-term future predictions. By incorporating physical equations, our model demonstrates temporal extrapolation and spatial generalization capabilities. Experimental results show that our approach achieves the state-of-the-art performance in spatiotemporal prediction across both numerical simulations and real-world extreme-precipitation nowcasting benchmarks.

Published as a conference paper at ICLR 2025
PINP: PHYSICS-INFORMET NEURAL PREDICTOR WITH LATENT ESTIMATION OF FLUID FLOWS
Huaguan Chen', Yang Liu?, Hao Sun
"aalling School of Artificial Intelligence. Renmin University of China, Beijing, China
Schooo of Engineering Science, University of Chinese Academy of Sciences, Beijing. China Emails: huaquancheng ruc. edu. cn; 1 iuyang22@ucas, ac. en; haosun un 8 ruc. edu.
ABSTRACT
Accurately predicting fluid dynamics and evolution has been long-standing chal- lenge in physical sciences. Conventional deep learning methods often rely on the nonlinear modeling capabilities of neural networks to establish mappings between past and future states, overlooking the fluid dynamics, or only modeling the ve- locity field, neglecting the coupling of multiple physical quantities. In this paper, we propose new physics-informed learning approach that incorporates coupled physical quantities into the prediction process to assist with forecasting. Central our method lies in the discretization of physical equations, which are directly integrated into the model architecture and loss function. This integration enables the model to provide robust, long-term future predictions. By incorporating phys- ical equations, our model demonstrates temporal extrapolation and spatial gener- alization capabilities. Experimental results show that our approach achieves the state-of-the-arr performance in spatiotemporal prediction across both numerical simulations and real-world extreme- precippatciiiitiiionn nowcasting benchmarks.
INSTRUCTION
The prediction and analysis of fluid dynamics play crucial role across many fields of science. Fluid phenomena span a wide range of scales, from macroscopic atmospheric circulation to microscopic intracellular transport, underscoring the complexity and significance of fluid motion. However, noisy data, experimental limitations, and the inaccessibility of physical quantities present substantial chal- lenges in learning the underlying dynamical systems and accurately predicting future fluid evolution.
These challenges primarily arise for two reasons. First, latent physical quantities are inherently difficult to obtain. For example, the advection and diffusion of substances within fluid flows governed by the Navier-Stokes (NS) equations, which couple multiple variables- as velocity, pressure, and concentration-innto complex, interdependent system. Capturing real-time data, like velocity and pressure, without interference is highly challenging. Successful approaches usually rely on supervised learning for these quantities, but such data is not easy obtain practical scenarios. While the HFM (Raissi et al., 2020) has made notable progress by inferring latent physical quantities from PDEs and observed concentration data, still falls short of predicting future fluid evolution.
Second, the complexity of the NS equations presents substantial obstacle. These nonlinear equa- tions couple multiple physical quantities, making the direct prediction future fluid behavior nearly impossible. A promising approach is the use of neural operators, which predict future physical fields based on past observations without explicitly solving the NS equations (Li et al., 2020; Tran et al., 2023; Wang et al., 2024). These models leverage the powerful nonlinear approximation capabilities of neural networks to map past fluid states to future. However, the absence of physical constraints often limits their interpretability and generalization ability. Another approach involves approximat- ing and simplifying the NS equations through methods like kernel methods or other theorems and applying neural networks estimate the fluid velocity field. By combining the estimated velocity field with simplified equations, these methods attempt to predict future fluid behavior (Deng et al., 2023: Xing et al., 2024). However, these methods lack intuitive clarity, and focusing solely on the velocity field overlooks the complex multivariable coupling intrinsic to the NS equations.
"Corresponding author

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Published as a conference paper at ICLR 2025
(a) Temporal extrapolation
Spatial generalization
(c) physical quantities
Training
Training
af (oos))
8(fe-s)
Tosting
Testing
c($p-s)
lu(tz 43)
Prodict
a(ta-s)
a(64-aa)
8(te- 4b)
Figure 1: Model has the following capabilities: (a) Temporal extrapolation, (b) Spatial generaliza- tion, (c) Latent physical quantities Estimation.
In this paper, we propose the Physics-Informen Neural Predictor, a new approach designed to address the challenges outlined above. For fluid dynamics prediction, discretize the NS equations in both time and space, incorporating them into model architecture and loss functions. Neural networks are employed establish a mapping function from past measurement (observed data) to both interme- diate observed quantities and latent physical variables, enabling predictions through the discretized equations. Unlike methods that solely establish mappings between past and future states focus only on velocity modeling, our approach estimates multiple physical quantities (e.g., velocity and pressure) to assist the prediction. To ensure the physical validity of the estimated physical quan- tities, we impose both physical and temporal constraints. By explicitly modeling physical quantities and incorporating the NS equations into the model, our method inherently possesses interpretability and enhances the extraction of latent fluid dynamics from observed data. This integration further improves the model's 's temporal extrapolation and spatial generalization capabilities in prediction.
To evaluate the effectiveness of the proposed method, conducted comparative study against several state-of-the-ae models for predicting future observed data. The evaluation was performed using both numerically simulated 2D and 3D datasets, as well real-world nowcasting benchmarks. The predictive accuracy of the models was quantitatively assessed across various scenarios.
Experimental results demonstrate that our approach achieves SOAT performance in predicting ob- served data, while simultaneously estimating multiple latent physical quantities for interpretability, and exhibiting better temporal extrapolation and spatial generalization capabilities (Figure 1).
In summary, our main technical contributions align to tackle the key challenges, given by:
Estimation of Multiple Physical Quantities and Assisted Prediction: We estimate latent physical quantities, which can assist in the prediction of observable physical quantities and serve as interpretable evidence to support our predictions.
2) Temporal Extrapolation and Spatial Generalization: Our model demonstrates the ability to extrapolate beyond the training steps and generalize across varying spatial domains.
Superior Performance: Our model consistently achieves SOTA performance across wide array of benchmark tests, including synthetic and real-world datasets (both 2D and 3D).
RELATED WORK
Modeling fluid flows has always been significant challenge. Since it is typically impossible to cal- culate explicit solutions for NS equations, scientists have developed numerous numerical methods to address this problem, such as the finite element method (Donea & Huerta, 2003), finite difference method (Godunov & Bohachevsky, 1959), finite volume method (Jasak, 1996), and spectral method (Orszag, 1979). However, these numerical methods heavily rely initial conditions, which are often difficult to obtain practical applications. Moreover, numerical methods face high computa- tional costs, especially when dealing with complex and variable scenarios and large computational domains. The complexity of modeling and the lengthy computation times render numerical meth- ods impractical in such cases. In recent years, the development of deep learning technologies has provided new possibilities for solving this problem. Based on whether latent physical quantities are modeled, we can broadly categorize these approaches into the following methods.
Neural Operator methods. This category of methods primarily leverages the powerful nonlinear modeling capabilities of neural networks approximate the complex mapping between inputs and
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outputs, thereby enabling predictions based past fluid data(Rao et al., 2023). For example, Deep- ONet (Lu et al., 2021a), which is derived from the universal approximation theorem, employs a branch-trunk architecture. FNO (Li et al., 2020) approximates the integral operator by employing linear transformations in the Fourier space. U-FNO (Wen et al., 2022) and U-NO (Rahman et al., 2022) enhance FNO with U-shaped multi-scale frameworks. Geo-FNO (Li et al., 2023b) handles complex geometries by transforming irregular input physical domains into latent spaces with uni- form grids. F-FNO (Tran et al., 2023) improves FNO through separable spectral layers and residual connections. Additionally, MWT (Gupta et al., 2021) introduces multiwavelet-base
neural oper- ator. LSM (Wu et al., 2023) decomposes complex nonlinear operators into multiple base operators, using neural spectral blocks to solve high-dimensional PDEs. However, these methods do not ex- plicitly model latent physical quantities, leading to a lack of physical interpretability. Overlooking the fluid dynamics, they perform poorly in temporal extrapolation and spatial generalization.
Latent physical quantity modeling methods. This category of methods based on existing NS equations and explicitly models the required physical quantities, such velocity, for fluid predic- tion. One approach is to directly incorporate the NS solutions as parameters of the deep model and formalize the constraints, e.g., PDEs, corresponding initial and observed data, loss function (Raissi et al., 2019; 2020; Lu et al., 2021b). enabling the inference latent physical quantities from observed data. However, this method is not effective for predicting future fluid behavior. Another approach is to approximate and simplify NS equations and estimate the fluid velocity field, then use the velocity field and the simplified equations to predict future fluid behavior. For instance, Zhang et al. (2022) initially provided estimates using an optical flow predictor constrained by the NS equations. DVP (Deng et al., 2023) combines vortex particles with a vortex-to-velocity dynam- ies mapping to capture complex flow dynamics. HelmFluid (Xing et al., 2024) integrates learned Helmholtz dynamics generate future fluid behavior. However, this method does not achieve ex- plicit incorporating of the NS equations, lacking intuitiveness. In addition, since fluid dynamics is a complex system involving the coupling of multiple physical quantities, models that rely solely on the velocity field have inherent limitations.
PHYSICS-INFORME NEURAL PREDICTOR
Let us consider a scenario: the transport of dye in water. The dye is transported and diffused with the flow of water, and it does not affect the flow of the water. Under normal temperature and pressure, water approximately satisfies the incompressible Navier-Stokes equations. We aim to predict the future evolution of the dye concentration only based on observed dye concentration data, while also estimating the unobserved velocity and pressure fields of the water as interpretable evidence.
Figure 2 illustrates the overall architecture of our PINP model. Herein, the dye flows from left to right through pipe with obstacles. The observed data includes dye concentration and spatial information v.. Using these observed data, and based on physies-informed neural network, we simultaneously estimate the concentration field at time t', as well as the underlying velocity and pressure fields which are unmeasured. With these inferred fields and the Discrete PDEs Prediction Network, we can predict the concentration field è' for the next time step.
In the Discrete PDEs Prediction Network, we construct the predictor using discretized PDEs. Since this discretization naturally introduces numerical errors, predicted concentration field, resulting in the final prediction a'. Additionally, the simultancously estimated velocity and pressure fields at time t' serve as interpretable evidence.
correction network to refine the
Notation.
Vx denotes spatial gradients, è = de time derivatives, V u (V)) divergence, and tensor inner product. k E (1.2....,N) ... denotes the k-th frame, 4 the time of k-th frame, and At the time interval between 4 and 14+1.
Throughout the paper, V =
de
3.1 DISCRETIZATION PDES
We considered the transport of a passive scalar c(x,t) E RxW by a flow velocity field u(æ, () E R2*H*W The passive scalar is advected by the flow and diffused but has no dynamical effect on the fluid motion itself, which satisfies the incompressible NS equations:
ö(x, 1) u(x, 1) Ve(xe, 1) Pe V -
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Physica-Informed
p(t)
y
Physical Inference
Neural Network
ru(t')
Discrete PDEs Prediction Network
<(s 1s).
c(te:s)
<(t')
(b) Discrete POEs Prediction Network
d(t+1)== = d(b.) () (-u(t)
Physical Constraint
p(t')
Constraint
(c(E)
Correction Network
df (fees)
o(ta41)
Temporal Constraint
<(e
c(fa)
Neural Network
Discrete PDEs Predictor
Constraint
Forward
Loss
Figure 2:
Schematic architecture of the proposed Physics-Infone Informed Neural Predictor (PINP).
where Pe denotes the Peclét number, c(2, t) the concentration of the passive scalar, which can be observed without interfering with the state of fluid flow.
According to the aforementioned equations, if we need to predict the concentration at future time points, we can derive the expression for the future concentration as follows:
c(ar, (++1) = ö(x;, l)dt + c(x,, Lx).
The integration term in Eq. 2 is undoubtedly extremely complex and challenging solve. However, considering the continuity of material concentration c(x, t) over time, based on the Lagrange Mean Value Theorem, there must exist moment between time tj and tx+1 such that:
ö(æ,, l)dt c(x;, D') i' [Lx, (x+)).
(3)
Therefore, the prediction process can be described by the following equation:
c(x, tx+1) ) = c(x (x) c(xx,')44
(4)
c<xx, (v) (-u t'). Ve(x,t") Pe -e- ve-'v'e(x,tt)) At.
Eq. is the final discrete form of the NS equations that we utilize.
3.2 PHYSICAL INFERENCE NEURAL NETWORK
Based on Eq. 4, we can predict the concentration. We need a way to model concentration data c(x, Ł'), velocity data u(a, f'), and the Peclét number (Pe) included in the equation. Considering the powerful nonlinear fitting capabilities neural networks, we utilize them to establish the mapping function from past K frames of data to the physical quantities at time t':
((x, t') = Fe, (c(x, S tx), v). t' E [tx, tx+1l:
(5)
where o denotes the physical quantities, c(x, < tx) = [c(x, (oc.(+11K))..... c(x, tx-1)) c(x, tx)] R H xW z E CCC the spatio embeddings.
e
For spatial information v. we define it as 10 = (a, d, b), where x 22x denotes the coordi- nates of grid points. For bounded fluids, d and b collectively characterize the internal positions and boundary information of the space. l W denotes the Signed Distance Function (SDF, Osher & Sethian, 1988), which describes the shortest distance from each position in the physical field to

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the obstacles. E IR H is used to describe various attributes of each position in the physical field, with different values of b representing boundaries and regions that permit fluid flow.
We set Pe learnable parameter in the network since the Pe is constant.
3.3 DISCRETE PDES PREDICTION NETWORK
Based on Eq. 4, we define the Discrete PDEs Predictor:
(x, tx+1) c(2r, (x) + (~u(x, Ve(a, (V)(x,( At.
(6)
Since the discretized PDE predictor may introduce errors, the predicted concentration field is cor- rected using a correction neural network (Gn):
0(x, (++1) (x,)), ((2,
3.4 SPECIFIC MODEL. IMPLEMENTATION
In Eq. 5, the mapping function need to establish is complex nonlinear function with coupled multiple physical quantities and multi-scale features. Since U-Net (Ronneherger et al., 2015) 5) is well- suited for modeling multi-scale information, 3D U-Net is considered more effective in capturing spatiotemporal features. Here, we adopt 3D U-Net (Çiçek et al., 2016) for the mapping:
Fog tz), v) F Na (c(a, S tz), v)).
Similarly, in Eq. 7, establish correction network using the U-Net architecture. Since the cor- rection task is relatively simple, the model used here is more lightweight. Appendix A presents more details, including the gradient operator, signed distance field, mask settings and model structure.
Go (é' (a, 11+1), c(x, t))
G U-Net (lghtweighn) (é' (xx 1x+1), c(x, tt)).
3.5
LOSS FUNCTION
Our loss function consists of three components: Data, Physical, and Temporal constraints. These components ensure the prediction accuracy and inference of underlying physical quantities.
Data constraints. Using the observed data (training labels), we impose constraints on the predicted values both before and after correction:
LDa (x, tx) = | (xx, tx) - c(x, fx 1 ||č(,, tx) - c(x, tx)
(10)
Physical constraints. We consider the physical constraints of the incompressible NS equations:
ú(æ 1) = u(x 1) Vu () Vp(x, + Re v2,(x,
(11)
V (( - 0.
Over very short time interval, we assume that uxx, l') u(x (x) and p(æ, (') ss p(æ, (x). We establish the residual terms by discretizing and non-dimensionaaiziing the aforementioned equations:
ei (xr (R) = Au(a, (x) + (-u uax (x) (2, 4x) - Vp(a, 4x) + Re V2u(x, (n)) At, c2(x, tu) = (V u(x At..
(12)
So, define the Physical constraint
Lphysicul (x, tx) |le 1 + (x, t4)l
(13)
Notably, in Eq. 12, we have introduced the pressure term p(æ,t') E #x and the Reynolds number Re. Similar to our approach for handling concentration and velocity, we utilize neural networks establish the mapping from the past observed data to the pressure. Re is treated learnalle parameter. In this way, we introduce H W + new variables while adding x x W constraint equations, further enhancing the physical constraints.
Temporal constraints. In Eq. 3, we have [tk. f11). However, directly imposing constraints on /' is both challenging and unnecessary. Instead, applying constraints directly on c(æ, t'). Over
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(a) Simulated Data
(b) Real World Data
Data
Spatial Resolution
Fluid 2D
640 256 256 x 256
Smoke 2D
Smoke 3D
61 x 64 x 64
SEVIR
96 x 96 (down sample)
Fluid Smoke 2D
Smoke 3D
SEVIR
Figure 3: Overview of the data with spatial resolution.
very short time interval, assuming steady flow, we can approximate the temporal evolution of c(2r, as monotonic bounded function. Since t' lies between la and 1x+1, we have:
||c( t¹) c(x, fx)ll 2 I [co(x,tx+1) c( tx)
(14)
Based on this, we establish the residual term as follows:
Temporal (ar, 1x) max max [1lc(1, f²) c(æ, tz) ||e(,, fA+1) c(ar, fx) 0}.
(15)
This implies that if ||c[x, t') c(x, fx)) 1 is less than or equal to ||e(æz fx+1) - c(2, tx. t.) we consider the concentration time term for this component is set to O. However, if [|c( (*) c(ær, (x)) is greater than ||c(ar, (x+1)
be optimized within an acceptable error range, and loss
c(x, 4x 112, we consider the concentration value time 1' to require further optimization.
In summary, our loss function assembles the above constraints and is written as:
N
1 NHW
C
(2., (Cratu (xx, (x) + LPhysical (,, (x) + Lremporal (at, (a)).
(16)
EXPERIMENTS
We extensively evaluate our model across four benchmarks, including both simulated and real-world scenarios, as well as 2D and 3D settings. The training and testing data consist solely of observable data. Overview of the data with spatial resolution can be found in Figure 3.
Table 1: Comparison of prediction performance on synthetic
4.1 SIMULATED DATA
data with test metrics of MSE (L) and MAE (4). For clarity, the best result is shown in bold and the second best is underlined.
Baselines. Given the simulated data, we compare our model with seven recognized advanced mod- els, including five baselines that are purely data-driven to approxi- mate complex mapping: FNO (Li et al., 2020), F-FNO (Tran et al., 2023), U-FNO (Wen et al., 2022), U-NO (Rahman et al., 2022) and LSM (Wu et al, 2023), and a base- line that utilizes established ve- locity fields to assist predictions: HelmFluid (Xing al., 2024), and two other baselines for vision
Flow 2D
Smoke 2D
Smoke 3D
Method
MAE
MSE
MAE
MAE
MSE
U-Net (2015)
0.0362 0.0076 0.0346 0.0140 0.1397 0.0385
ResNet (2016)
0.0424 0.0092 0.0412 0.0192 0,1460 0.0419
FNO (2020)
0.0300 0.0029 0.0303 0.0118 0,1465 0.0413
U-NO (2022)
0.0183 0.0008 0.0284 0.0092 0.1386 0.0363
U-FNO (2022)
0.0187 0.0011 0.0335 0.0125 0.1237 0.0314
F-FNO (2023)
0.0336 0.0055 0.0282 0.0107 0.1124 0.0250
LSM (2023)
0.0338 0.0037 0.0311 0.0109 0,1193 0.0293
HelmFluid (2024)
0.1222 0.0503 0.0254 0.0085
0.1217 0.0298
PINP (this work) 0.0107 0.0003 0.0209 0.0057 0.1073 0.0249
tasks: U-Net (Ronneberger al., 2015) and ResNet (He et al., 2016).
Fluid 2D. We consider scenario where a substance is transported by fluid flow through pipe. An incompressible fluid with a certain concentration flows from left to right, passing through multiple solid objects with varying positions and shapes. The substance is advected and diffuses with the fluid, and its concentration is the observed data. The real-world scenario involves the transport of dye carried by water flow through pipe. To evaluate the model's temporal extrapolation and spatial generalization capabilities, the training process involves predicting the next 5 frames of concentra- tion based the previous 4 frames. During testing, the model is challenged to predict 40 future frames concentration from the 4-frame input. Additionally, the test set includes variations such the addition, removal and shape changes of obstacles, which
present in training set.

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(a) FlowZd
(b) Smoke2d
(c) Smoke3d
0.05
MSE (log
-3-
0.02
MSE
0.03-
0.00
0.01
20
40
75
160
2.5
7.5
16.0
Frame
Frame
Frame
FNO
UNO
UFNO
LSM
HelmFluid
Figure 4: Comparison of MSE for different models on each prediction frame.
(a) Concentration prediction comparison
(b) Latent physical quantities Prediction
Real 40)
U-NO
Prediction Error
weas
Prediction Error
Figure 5: (a) A comparison the 40th frame prediction results between our method and U-NO, U-FNO. (b) The inferred and predicted velocity and pressure fields the 40th frame.
As shown in Table 1, PINP significantly outperforms other models, achieving a notable reduction in emror compared the second-best model (MAB: 0.0107 0.0183). In the frame-by-frame fluid prediction comparison (Figure 4 (a)), our model also outperforms others.
To further illustrate the effectiveness of model, Figure (a) presents several examples. Com- pared to U-NO and U-FNO, our model predicts fluid evolution more accurately, particularly in regions where the substance interacts with boundaries. Additionally, our model estimates velocity and pressure fields at each prediction step, providing interpretable evidence that supports the predic- tions and highlights its advantages in capturing complex dynamics. Figure 5 (b) shows the estimated latent physical quantities at the 40th predicted frame. The inferred physical quantities have physical significance, including the velocity distribution at the edges of obstacles and the pressure changes before and after the obstacles. These quantities can serve as interpretable evidence for predictions.
In Figure 4 (a), some of the baseline models exhibit periodic oscillations in the MSE as the number of prediction frames increases. This phenomenon is quite unusual because, under normal circum- stances, a well-trained model should not exhibit such erratic oscillations. This might be due to rollout training with small number of steps. We have discussed this issue in Supplementary Material.
Smoke 2D. Smoke rises and diffuses in an enclosed space, with smoke concentration as the ob- servable data. During training, the model predicts the next 10 frames of concentration based on the previous 4 frames. During testing, the model need to predict 100 future frames based on the first frames input, with variations in smoke source locations between the training and testing sets.
In the 2D smoke concentration prediction task, PINP significantly outperforms other models (Table 1). In the frame-by-frame prediction comparison (Figure 4 (b)), both HelmFluid and U-NO ini- tially perform better than our model. But their prediction errors increase rapidly as the prediction progresses. This demonstrates the temporal extrapolation capability of our model.
Figure (a) presents an example showing that, compared to U-NO and HelmFluid, our model pre- dicts the movement of smoke more accurately, particularly the shape of the smoke plume and the direction of the smoke column. Additionally, Figure 6 (b) shows comparison between our inferred and predicted latent physical quantities and the ground truth in the smoke region. The velocity and pressure fields, estimated solely from observable data, resemble the true values, effectively serving as evidence for the interpretability of our method.

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Concentration prediction and comparison
(b) Latent physical quantities Prediction
(c) Correlation
Real
U-NO
HelmFluid
PINP
Me
50
Real
Correlation
Myy
0.9039
Predictin
Error
Pred.
K
0.9099
0.9251
100
100
Real
L-
100
Correlation
Prediciion
M4y
0,9207
Error
Pred.
Figure 6: (a) A comparison of prediction results between method and U-NO, Helmfluid. (b) A comparison between our predicted latent physical quantities and the ground truth and (c) Correlation.
As shown in Figure 6 (c), high cor- relation (2 0.9) demonstrates the sim- ilarity of our estimated latent physical quantities with the true data, indicat- ing that the estimated physical quanti- ties contribute to understanding the fu- ture motion process. However, it is im- portant note that, due to the unknown physical scale, scaling relationship ex-
HelmFluid Real
150
125
100
ists between the estimated latent physi-
PINP
cal quantities and their true values. To obtain the values of these estimated quantities, the initial values of the real physical quantities or any real data point are necessary to determine the scaling constant. A detailed discussion of this is provided in Appendix D.
Figure A comparison between our method and Helm- fluid for 3D smoke prediction, with the predicted frames increasing from left to right.
Smoke 3D. Similar to the 2D smoke scenario, we extend our method to 3D setting. During training, the model predicts the future 4 frames of concentration based the past frames. During testing, model is required to predict the future 10 frames concentration using the past frames, with varying smoke source locations and concentrations between the training and testing sets.
In the 3D smoke concentration prediction task. PINP significantly outperforms other models (Table 1). In the frame-by-free -ramee prediction comparison (Figure 4 (c)), model also outperforms others. Figure presents an example. Compared to Helmfluid, our model preserves more detailed features. The prediction results the top surface significantly better than those predicted by Helmfluid.
4.2 REAL-WORLD DATA
We consider the nowcasting prediction problem in real scenarios, which can be viewed as the trans- port problem of water-bearing clouds driven by wind. Unlike the simulated data, nowcasting data is boundary-less and contains noise. It is important to note that the baseline from the simulated data test is not applied to this task, and we establish new baselines for this problem.
Baselines. The baselines for nowcasting include video prediction models: ConvLSTM (Shi et al., 2015), PredRNN (Wang et al., 2022), Si (Gao et al., 2022a) and Earthformer (Gao et al., 2022b). And physics-driven prediction model: NowcastNer (Zhang et al., 2023). NowcastNer uses de- terministic prediction model (Evolution network) paired with a probabilistic model (GAN). The Evolution network outperforms the GAN-based model in most metrics. Results for the Evolution network are shown Table 2, while full comparisons of NowcastNet are Appendix (Table S1).
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Table 2: Comparison of prediction performance on SEVIR. The test metrics are MSE(() and CSI(T). For clarity, the best result is shown in bold and the second best result is underlined.
Method
MSE (1e-3)
CSI-M
CSI-219
CSI-181
CSI-160
CSI-133
CSI-74
CSI-16
CoM (2015)
4.0082
0.3490
0.0340
0.1441
0.2003
0.3274
0.6574
0.7293
PredRNN (2022)
4.0383
0.3382
0.0155
0.1203
0.1880
0.3252
0.6551
0.7283
SimVP (2022a)
3,8508
0.3567
0.0709
0.1496
0.1997
0.3212
0.6562
0.7424
Earthformer (2022b)
4,4029
0,3585
0.0661
0.1663
0.2163
0.3400
0.6428
0.7196
NowcastNet (2023)
3.8883
0.3743
0.0803
0.1811
0.2351
0.3515
0.6635
0.7339
PINP (this work)
4.1684
0.3800
0.0989
0.1952
0.2448
0.3573
0.6556
0.7279
The Storm EVent ImageRy (SE- VIR) dataset (Veillette et al., 2020) con- tains metcorological events across the United States between 2017 and 2019.
SEVIR.
min
20 min
min
40 min
min
60 min
PredKNY CORYLSIM Real
the NEXRAD radar composite of Vertically Integrated Liquid (VIL), which covers area of 384 km x 384 km with a spatial resolution of km and temporal interval of minutes. Following Gao et al. (2022b), we evaluate nowcasting by predicting VII. for up to minutes (12 frames) based on 65 minutes of past VII data (13 frames). We down- sample the resolution to 96 x 96.
For nowcasting,
SiaV P
NowCCeNNN Earthformer
As shown Table 2, while our model does not lead in MSE or low CSI met- rics, achieves the best results in the aver-
age CSI and high CSI metrics, highlight- ing its potential in extreme precipitation prediction. Figure presents an compari- son of our model against other baselines, In nowcasting, the NS equations are not strictly satisfied. This test evaluates the model's 's predictive performance and ro- bustness to noise under conditions where these equations are not fully obeyed.
PINP
00-1
16-31
31-59
59-74
74-100
100-133
133-160
160-181
181-219
219-255
Figure 8: Comparison of results on the SEVIR dataset.
Appendix B presents more details, including the generation methods, data scale, details of SEVIR and the evaluation metrics for the test results. Details regarding the training setup, hyperparameter configuration, and multi-loss function training can be found in Appendix C. Further baseline results and physical field inference results can be found Appendix F.
5 ABLATION STUDIES
Table 3: Ablation Experiment Setup and Results
Ablation Studies Setting
MAE
MSB
We conduct ablation experi- ments on constraints, model components, and the discrete PDEs Predictor. Results are shown in Table 3, while Figure 9 illustrates the change in MSE with the frame under different ablation experiments. The leg- ends (e.g., C1, MI, D1) in Fig- ure 9 correspond the ablation case labels in Table 3. Details
C1. Normal
0.0209
0.0057
C2, no Physical Constraint
0,0272
0.0089
Constraint
C3. no Temporal Constraint
0.0293
0.0108
C4. Velocity-Presssue -Pressue Constraint
0.0262
0.0087
M1. Normal
0.0209 0.0057
Model
M2. no Correction Network
0.0316
0.0098
DI. Normal
0,0107
0.0003
Discrete PDEs
D2. changing Discrete PDEs
0.0197
0.0010
D3. replacing c(t') with c(tx)
Inf
Inf
for these ablation experiments can be found in Appendix F.

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(a) Constraints
(b) Model
(c) Discrete PDEs
0.025
0.0020-
0.020
-22
0.020
0.0015
D3
0.015
MSE
MSE
0.015
MSE
0.0010-
0.010
0.010
0.0005
0.005
M1
0.000
0.000
0.0000
25
50
75
100
0
50
100
10
20
30
40
Frame
Frame
Frame
Figure 9: Ablation experiments: (a) constraint, (b) model, and (c) discrete PDEs.
Constraint Ablation. In method, We introduced multiple constraints. We performed experi- ments by removing the physical and temporal constraints to evaluate the impact on performance. Besides, validate whether the inclusion of pressure p beneficial for prediction, an experiment: no Velocity-Pressure Constraint, which removes é₁ in Eq. 12. Results in Figure (a) and Table 3 show that the inclusion of constraints and p is beneficial for prediction.
also designed
Model Ablation. We introduced a correction network to refine the gradient operator. We conducted experiments evaluate performance without this network. The results are shown in Figure 9 (b). The correction network notably improves accuracy, especially in later predicted frames.
Discrete PDEs Ablation. In Eq. 4, the prediction operator use c(t') to predict, which might be re- placed with c(fx). We conducted two sets of experiments: one where c(t') replace c(ta) and another involving modifications to the prediction operator. The results, presented in Figure 9 (c), show that while these adjustments improve accuracy in the initial frames, the error increases significantly as the number of predicted frames increases.
Figure 10 illustrates our model's performance during testing alongside its memory consumption. Our model employs 3D structure in both 2D and 3D scenarios, this results in higher memory usage in 2D scenes. Con- versely, in 3D scenes, our model exhibits relatively lower memory usage.
2D Model
3D Model
0,030
MAE
0.025
0.12
0.11-
0.020
1000
1500
2000
1000
2000
3000
4000
5000
Better
Memory Consumption (Mb)
Momory Consumption (Mb)
FNO
UNO
UFNO
LSM
HelmFluid
PINP
Figure 10: Comparison of GPU memory consumption and MAE
CONCLUSION
This paper introduces the Physics-Infommnn Informed Neural Predictor (PINP) model, designed for predicting spatiotemporal evolution. The PINP model leverages latent multi-quantity modeling and integrates PDEs directly into the neural network framework, enabling enhanced understanding and accurate prediction of dynamical systems.
key feature of the PINP model its ability estimate latent physical quantities that are not directly observed in the data. This provides more comprehensive analysis of the physical processes underlying the observed dynamics. The model's architecture embeds the governing PDEs, ensuring that physical constraints are adhered to during both training and prediction phases. This integration leads to improved consistency with physics, facilitating robust long-term forecasts.
When evaluated against range of baseline methods, the PINP demonstrates better prediction curacy, particularly in its ability to extrapolate over extended temporal horizons and generalize spa- tially. This confirmed through testing multiple benchmark datasets, where PINP's predictions aligned more closely with ground truth than those of competing approaches. future research, we aim expand the application of the PINP model to more complex and comprehensive tasks, ac- counting for other type of governing PDEs.
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ACKNOWLEDGMENT
The work is supported by the National Natural Science Foundation of China (No. 92270118 and No. 62276269), the Beijing Natural Science Foundation (No. 1232009), and the Strategic Priority Research Program of the Chinese Academy Sciences (No. XDB0620103). In addition, H.S and Y.L. would like to acknowledge the support from the Fundamental Research Funds for the Central Universities (No. 202230265 and No. E2EG2202X2). Our source codes are available following GitHub repository: https: //gp::://iiie thub. com/ intelneellll scci comput/PINg
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APPENDIX
A IMPLEMENTATION DETAILS
In this section, we provide additional details on key aspects of the paper, including the Gradient Operator, Signed Distance Field (SDF), Mask Settings and Model Structure.
For the gradient operator, we discuss its approximation and the associated errors, particularly in fluid-boundary interactions. To address these issues, we introduce specific masks-Maski for the first-order gradient operator and Mask2 for the second-order operator-to reduce these errors.
The Signed Distance Field (SDF) helps define boundary points and enhances spatial understanding. which is crucial for handling irregular shapes and obstacles. The SDF also plays key role in the boundary point identification process (Figure S1).
Considering that the established mapping function in Eq. 8 is nonlinear and multiscale, archi- tecture employs a structure similar to 3D U-Net to build complex mappings and multiscale features.
These detailed approaches contribute to improving the accuracy and robustness of the overall pre- diction model.
A.1 GRADIENT OPERATOR
Using the Navier-Stokes equations, transform the partial derivative of concentration with respect to time into spatial gradient terms. For these gradient terms, we employ the second-orde central difference method to achieve estimation. Letting be an interior point with x Ax and + Ar be points neighboring it to the left and right respectively:
(x)
T (51)
(x + Ar) = f(x) + (x) +
2 Ar
E1 (I, Ax)
6
(S1)
(x)
T (Ea)
Ar) =
Ar f(x)
A..
A
E2 E (x, - Ar).
2
Solving the linear system, we derive:
f()) f + Ar) - Az)
(S2)
2Ax
To achieve better optimization results, we exclude gradients of boundary points from the training loss calculation.
A.2 SIGNED DISTANCE FUNCTION
A signed distance field (SDF) measures the orthogonal distance from given point in metric space to the boundary
set Q with the sign determined by the location of æ relative to Q. Specif- ically: If is inside Ω the signed distance is positive. The distance decreases boundary of Ω. If x: is outside Ω the signed distance becomes negative. essence, the signed distance field provides both magnitude of the shortest distance the boundary of the and sign indicating whether the point is inside or outside the boundary.
approaches the
d(x) min(d|a, boundary))
((x) = 0, z E (æld(a) M, Vz)
Figure S1: An example of Signed distance function and Mask
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2
4
mask 1
2
4
mask2
2
4
2
4
V2
v5
V10
V17
2
2
2v
v13
2v5
3
3
V10
V13
3v2
4
V17
2v5
4v2
Figure S2: Maskl and Mask2.
In this paper, we define Q as the set of external boundarics and obstacle boundaries. Since no fluid flows through the obstacles, do not assign negative values to the points inside the obstacles; instead, we set them to zero.
In addition providing richer spatial information, the directed distance field helps identify boundary points. As mentioned earlier, to achieve better optimization results, we exclude the gra- dients of boundary points from the training loss calculation. While boundary points the external boundary are easy to identify, determining the boundary points for internal obstacles, especially those with irregular shapes, is more challenging. This problem can be addressed using the directed distance field. We define the orthogonal distance from point x to the boundary of set S2 as d. Points with a distance less than M from the boundary are considered boundary points. Therefore, the set of boundary points is given by {a (r() < M, Vr}.
Figure S1 illustrates an example, including the definition of distance(left) and the setting of the loss at boundary points to zero(right).
A.3 MASK SETTINGS
The specific boundary handling illustrated Figure S2. Since the gradient operator in Formula S2 introduces approximations, these approximation errors become more pronounced during fluid- boundary interactions. To reduce such errors, we employ a masking approach. For first-order gradi- ent operators, define Maskl:
Mask1 := (x<|d<>)) S 2.5, VrJ.
(S3)
and for second-order gradient operators, we define Mask2:
Mask2 := {x|d(<) < 3.5, Vr).
(S4)
This helps ensure better handling of boundary interactions and minimizes the impact approxima- tion enrors in fluid simulations.
A.4 MODEL STRUCTURE
The model adopts a recurrent prediction structure, utilizing a sliding window approach for fore- casting. Each time, it predicts one frame based the previous four frames and incorporates the predicted frame into the window for subsequent predictions. The prediction architecture for each step is illustrated in Figure 2.
In Figure 2, we use Physical Inference Neural Network to establish the mapping function from past data to the physical quantities at time L'. Considering that the established mapping function is
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Concat
11,1
Linear
dst), t)
Linear
[B, (.W.W)
3D Conv Block
Permute
Linear
Permute
3D Conv Block
(8,48,T,H,W
Conv Block
(B, 24, H.W)
Conv Block
Spatial Up
Spatial Down
3D Conv Block
Batch Norm3d
3x3x3
Spectral Norm
Conv
3D Block
H/2.W22
Batch Normád
3x3=3
Spectral
3D Conv Block
ReL.U
48, T, 1/2,W/2)
Conv
Spatial Up
Spatial Down
Block
3D Conv Block
(B, 192, T, H/4,W4)
Spatial
3D Conv Block
(B, 96, I, H/4,
MaxPool3d (1.
Spatial Up
Spatial Down
Block
3D Conv Block
(B, 384, H/8.W8)
Spatial Up
(B. 192, H/8.W8))
3D Conv Block
Upsample (1.2.2)
Spatial Down
Spatial Up
3D Conv Block
384, T, H/16,W/16)
Figure S3: Detailed structure of our model, using the complex mapping from c(s (), db to ul(() as an example.
nonlinear and multiscale, our architecture employs a structure similar to 3D U-Net to build complex mappings and multiscale features.
We present the detailed structure of our model, using the complex mapping from c(S (), 1/ uu(') as an example (Figure S3). The structures the remaining mapping functions are similar to this.
A.5 NOWCASTNET SETTING AND RESULT
is important to note that NowcastNet utilizes a deterministic prediction model (Evolution network) combined with probabilistic generative model (GAN). Given that this is deterministic prediction task, the Evolution network outperforms the combined generative model (GAN) in most evaluation metrics. In the main text, selected the best prediction results from the deterministic model. In this section, we provide the complete results.
B DATA AND EVALUATION
In this section, we provide a detailed explanation the data and evaluation metrics, including details such the data generation method, data scale, and the specifies of the prediction tasks.
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Table S1: Comparison of NowCastNet results under different settings with Our Model
Method
MSE (Ic-3)
CSI-M
CSI-219
CSI-181
CSI-160
CSI-133
CSI-74
CSI-16
PySTEPS (2019)
8,4682
0,3373
0.0912
0.1756
0.2162
0.3244
0.5848
0.6313
Evolution network
(wrap mode = 'bilincar')
3.8883
0.3743
0.0803
0.1811
0.2351
0.3515
0.6635
0.7339
Evolution network
(wrap mode 'nearest')
6.1783
0.3529
0.0971
0.18
0.2220
0.3110
0.5986
0.7027
NowcastNet (2023)
5.7999
0.3549
0.0933
0.1844
0.2179
0.3104
0.6131
0.7106
PINP (this work)
4.1684
0.3800
0.0989
0.1952
0.2448
0.3573
0.6556
0.7279
Table S2: Data description includes spatial resolution, number of training samples, and train- ing/testing setup:
Data
Spatial Resolution
Training datasets
Training
Testing
Flow 2D
640 256 256 256 64 x 64 x 64
2500
4 to
4 to 40
Smoke 2D
1500
4 to 10
4 100
Smoke 3D
340
to
4 to 10
SEVIR
96 x 96 (down)
35718
13 to 12
13 to 12
B.1 DATASET
Fluid 2D. In this study, the dataset used consists of image sequences of fluid-driven mass diffu- sion, generated using the Computational Fluid Dynamics (CFD) method through COMSOL, soft- ware (Multiphysics, 1998). The physical setup involves an active flow field of width W and height H. filled with fluid of density p and viscosity coefficient jr. At the left side of the field the input, where a material with diffusion coefficient D. and concentration flows and diffuses, driven by an initial velocity u. The temperature is set to standard conditions, and the pressure is one atmo- sphere. The data used in the experiments consists only of concentration data. The parameters for data generation are shown in Figure $4.
This dataset, generated by COMSOL, is used for training and evaluating the model. The training dataset consists five subsets, each containing 500 concentration images representing the diffusion of materials in the fluid. These images form a time series, with each image having resolution of 640 x 256 pixels. The test dataset is divided into seven subsets, each containing 200 images of the same resolution, also forming time series data. Differences in physical field parameters across the subsets are primarily reflected in changes in obstacle positions.
To evaluate the spatial generalization ability of the model, the set includes three special subsets: one involves changes in the shape of obstacles, another simulates the removal of an obstacle, and the third adds a new obstacle the original physical field. These conditions are absent in the training data to assess the model's generalization performance and to determine if it has successfully learned the underlying physics.
To evaluate the model's temporal extrapolation ability, we used strategy during training where the model predicts the subsequent four frames based on the first four frames. the testing phase, the model is tasked with predicting up to 40 frames using the first four frames.
Smoke 2D. Compared to the fluid dataset, the smoke dataset features more complex scenes. We focus on the process of smoke rising and diffusing in the air, with square boundary defining the space. The data was generated using &Flow (Holl & Thuerey, 2024), with each image having a resolution of 256 x 256 pixels, and the location of the smoke source varies between the training and test sets. The data used in the experiments consists only of concentration data.
Similar to the fluid dataset, we strategy where the model predicts the next ten frames based on the first four frames during the training phase. In the testing phase, the model is required to predict up to 100 frames using the initial four frames. Different data has varying smoke source location.
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W 2.2m
Gy(H y)
0
[m/s]
H²
P x 10" (kg/ (kg/m²)
T 293.15[K]
6y(H y)
C
H²
(mol/m²)
μ [Pa..
p= 1[amm]
D. 1 10 m²/s]
0.88m
Figure S4: The configuration chart for data generation settings,
Smoke 3D. We extend the 2D smoke dataset to a 3D setting, considering the process of smoke ascending and expanding in a confined 3D space with a resolution 64 x x 64. The boundary is defined as a cube, and the location of the smoke source varies between the training and test sets. The data used the experiments consists only of concentration data.
Similar to the fluid dataset, we use strategy where the model predicts the next four frames based on the initial four frames during the training phase. In the testing phase, the model is required to predict up 10 frames using the first four frames. We used FactFormer (Li et al., 2023a) generate 340 sets of data for training. Different data has varying smoke source location.
SEVIR. Storm Event Imagery (SEVIR) (Veillette al., 2020) is dataset encompassing thousands of meteorological events, including various storms, lightning strikes, and precipitation events the United States from 2017 to 2019. Each event is documented with image sequences covering a 4- hour period over a 384 km x 384 km area. The dataset includes data from five sensors: satellite imagery, infrared (water vapor), infrared (window), NEXRAD radar composites of Vertically Inte- grated Liquid (VIL.), and lightning data. All sensors provide data with a spatial coverage 384 km 384 km and temporal resolution of 5 minutes.
For nowcasting, we selected the NEXRAD radar VIL composites. The VIL data has a spatial reso- lution of 1 km and is recorded 5-minute intervals. Following the approach of Gao al. (2022b), we use 65 minutes of VIL data (13 frames) to predict up to 60 minutes ahead (12 frames) for pre- cipitation nowcasting. Due computational limitations, downsample the spatial resolution to 96 x 96. This downsampled dataset includes 35,718 training samples, 9,060 validation samples, and 12, test samples.
The data overview is presented in Table $2, including spatial resolution, number of training samples, and training and testing setup.
B.2 EVALUATION METRICS
In the simulated data evaluation, MSE and MAE are used the evaluation indexes. In the real data evaluation, for the adjacent precipitation prediction index, we follow Veillette et al. (2020), and use the Critical Success Index (CST) to evaluate the prediction quality.
Given predictions (Tn.e.h.w) and corresponding ground truth (In,,,h,w)) Fn knathhw E IR, 1, ...., N, L...., ,,. I,..., H, w = L...,, W, the above-mentioned metries can be calculated as follows:
W ΣΣΣΣ
N
H
MSE
NTHW
n=l t=
h= ΣΣΣΣ
(S5)
1
MAE
Ma Inashalll
NTHW
For the inferred and predicted latent physical quantities, referring to Kochkov et al. (2021), we quantitatively evaluate them by calculating the Correlation.
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Table S3: Hyperparameter and Training setup
Type
Name
Mcaning
Value
Hyperparameter
Mask1
First-order gradient operator mask
Eq. S3
Hyperparameter
Mask2
Second-order gradient operator mask
Eq. S4
Training setup
LR
Learning Rate
1c-3
Training setup
Epoch
Number of training epochs
Training setup
Optimizer
Type of model optimizer
Adam
Training setup
Scheduler Batch Size
Schedule the learning rate of the optimizer the number of samples processed together
StepLR
Training setup
(2, 2, 32)
0": Value different for each dataset, the order of 2D fluid, 2D smoke, 3D smoke, and SEVIR.
E
(zn,th,w En (Em,e, In,t.h,.))
Correlation
(S6)
DE=1
V (Engehww Fnith,
Specifically, Veillette et al. (2020) used six precipitation thresholds which correspond to pixel values [219, 181, 160, 133, 74, 16). The prediction Fn,t,h. and the ground-truth In,t, are rescaled back to the range 0 - 255.
En w 2 T kn,th 2 D} #Hits(T)) = # (,, # #Misses(T) # (in,,,h, h,w
tn,,,hw 2 kn,1hh, 7}
(S7)
# F.Alarms(() = ( (m, h,
Ln,,, < T, In,thhw 2 T)
(Tits(T)
CSI T= #Hits(r) + # Misses( (T) + #F.Alarms(T)
(S8)
CSI-M Σ CSI- T, - TE 2219, 181, 160, 133, 74, 16).
# represents the number elements in the set, 7 E 2219, 181, 160, 133, 74, 16] is one of the thresh- olds. We denote the average CSI- over the threshoids [219, 181, 160, 133, 74, 16] as CSI-M.
must be stated that the extreme point of MSE and the extreme point of CSI are usually not consis- tent, and the model with the smallest MSE the validation set is selected for testing.
C TRAINING DETIALS
In this section, present the training details of our experiments, including hyperparameter settings, training setup, and the specifics of multi-loss training.
C.1 HYPERPARAMETERS AND TRAINING SETUP
The hyperparameters and experimental setup used in our experiments are shown in the table below:
C.2 MULTI-LOSS FUNCTION TRAINING
Since the model training process involves multi-loss optimization problem (Eq. 16), further show the training details.
Figure S5 (left) shows the change of Loss with the number of training rounds during training, and Figure S5 (right) shows the change of validation metrics with the number of training rounds.
As shown in Figure S5 (left) and Figure S5 (right), the loss corresponding the data constraint mains the largest, indicating that despite the introduction of multiple loss functions, the other losses do not significantly affect the primary constraint. Additionally, the loss associated with the temporal
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Main Loss
Temooral Loss
Main Loss2
validation
-1
0.0100
MSE (log 10)
Physical Loss
Tamporal Loss
0.0075
-2
MSE
0.0025
0.0000
10
20
10
30
40
10
20
30
Epoch
Figure S5: Training loss versus training epochs (left), loss of validation versus training epochs (middle) and Temporal loss versus training epochs (right).
Table S4: Correlation between Predicted and True Latent Physical Quantities.
Latent Physical Quantities
Correlation (PINP)
Correlation (HelmFluid)
Ug
0,9017
0.8891
Myy
0,9109
0.8997
0.8984
constraint reaches a value of zero, demonstrating that we can effectively bound the temporal loss in ||e(,, 1) c(x, 2) preventing it from degrading into:
Tempporl (æx, t) ||c(x, t') c(x,t))l t)l
(S9)
As shown in Figure S5 (middle), the test metrics validation decrease rapidly with the increase in training epochs and then level off. In sense, the physical constraints and temporal constraints as form of regularization, providing some resistance to overfitting.
D DISCUSSION ON LATENT PHYSICAL QUANTITIES
In this section, we will compare our estimated latent physical quantities with the baselines and provide detailed explanation.
Since the HelmFluid model capable of estimating the velocity field (but not the pressure field), we considered it relevant comparison for method. The comparative results of correlation the estimated latent physical quantities are shown in Table S4, We can see that our estimation subpasses that of HelmFluid.
Additionally, it's important to note that due to the unknown physical scale (since only the grayscale video sequences of the concentration field are provided as the training data), our estimated velocity fields and the true values have scaling relationship, expressed as follows:
(S10)
where o is scaling constant related the actual physical scale. When only concentration videos available, we cannot determine the exact physical scale of the region represented in the video (e.g.. the scale may lie in any range such as from 10 em I m in the smoke dataset). This uncertainty introduces a constant factor difference between the predicted and true values.
Thus, directly computing the MSE might be inappropriate. To address this, considered two methods for calibration:
Case 1: Using the true initial velocity values (assumed known a priori) to compute a.
Case 2: Using just one true data point to estimate a.
We then apply the computed a to correct the estimated results (note that HelmFluid also requires calibration). The calibrated MSE values shown in Table $5.
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Table S5: Comparison of calibrated MSE for latent physical quantities between model and HelmFluid.
Latent Physical Quantities
PINP (Case
PINP (Case 2)
HelmFluid (Case
HelmFluid (Case
0.1056 0.0806
0.3128
0.1482
0.5035 0.3407
By p
0.2806
0.1187
0.1539
0.4653
Table S6: Comparison of our model's 's results with those obtained after removing the physical con- straints and temporal constraints on 2D fluid data.
Metries no Physical Constraints no Temporal Constraints
no Selocity-Presssuee Constrain
Normal
MAE
0.0209
0.0272
0.0293
0.0262
MSE
0.0057
0.0089
0.0108
0.0087
The results in Tables S4 and S5 highlight the superiority of approach in predicting the latent physical quantities. Moreover, even with constant factor difference, high correlation (2 0.9) sufficient to demonstrate the consistency between our predictions and the true data in terms of distribution, regardless of scaling. This can serve as supplementary information for concentration prediction and provide interpretable evidence for the future evolution fluid dynamics.
E ABLATION STUDY
In this section, we conduct ablation studies to assess the contributions of different components of method, including constraint ablation, model ablation, and prediction operator ablation.
E. CONSTRAINTS
The constraint ablation experiment consists of two parts: removing the physical constraints and removing the temporal constraints:
Physical Constraints. We removed the physical constraints while retaining the temporal constraints and retrained the model. The results show a decline in prediction accuracy due to the absence of physical constraints (Table S6), and the inferred latent physical quantities exhibit issues, lacking real physical significance (Figure S6).
Velocity-Pressure Constraint.
To validate whether the inclusion of pressure beneficial for prediction, we also designed an experiment: no Eq. 12. Results in Figure (a) and Table show that the inclusion of constraints and p is beneficial for prediction. In Figure S6, we present this result, where it can be seen that the predicted velocity field is not correct.
Constraint, which removes e₁ in
Temporal Constraints. We removed the temporal constraints while retaining the physical con- straints and retrained the model. The results indicate that the prediction accuracy also declined due to the lack of temporal constraints (Table S6), and the inference of latent physical quantities, especially the pressure field, was problematic (Figure S6).
E.2 MODELS
In this section, we perform ablation experiments on our model structure by removing the correction network to verify its necessity.
Correction Network. As shown in Table S7, after removing the correction network, prediction accuracy significantly decreased. Figure S7 illustrates example where "stripes" appeared in the output after the correction network was removed. These artifacts are caused by errors in the gradient operator. By using the correction network, this issue can be effectively mitigated.
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Real 40)
Byy
Without
Physical
Constraints
Without
Temporal
Constraints
Without Velocity-
Pressure
Constrain
Normal
Figure S6: An example comparing the results of our model with those obtained after removing the physical constraints and temporal constraints.
Table S7: Comparison of our model's results with those obtained after removing the Correction Network on smoke data.
Metrics
Without Correction Network
Normal
MAE
0.0316
0.0209
MSE
0. 0.0098
0.0057
E.3 DISCRETE PDES
In this section, we conduct ablation experiments on the discrete PDE operators, including modifying the discrete PDE operators and replacing c(t') with c(t).
Table S8: Comparison of our model's results with those obtained after changing Discrete PDEs and replacing e(t') with c(t) on 2D smoke data.
Metrics
Change Discrete PDEs
Replacing e(t') with c(t)
Normal
MAE
0.0197
Inf
0.0107
MSE
0.0010
Inf
0.0003
Discrete PDEs. We modified the discrete PDE operators, with the original operator:
(x, c(x,t) u(x t') Ve(a, Pe Pe='vz((z,t')) At,
(S11)
and the modified version:
d (a. L + c(xx,) + u,( + +u (x, t').
(S12)
The comparison of results before and after the modification is shown in Table S8, where the predic- tion accuracy significantly decreased after the modification.
Replacing e(t') with c(t). In the prediction operator, use the physical quantities at time l' along with the discrete PDE prediction operator for forecasting, constraining the observable data between t and + 1. If we directly results, as shown in Table S8, indicate that errors accumulate rapidly when predicting beyond the training time steps. Figure S8 show example.
the observable data at time t guide the prediction, the experimental
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100
Real
Without Correction
Network
Normal
Figure S7: An example comparing the results of our model with those obtained after removing the Correction Network.
Table S9: Comparison of Evaluation Metrics the SEVIR dataset under Different Physical Loss Configurations
Method
MSE (Ie-3)
CSI-M
CSI-219
CSI-181
CSI-160
CSI-133
CSI-74
CSI-16
Lowered Physical Loss Weight
4.0305
0.3662
0.0781
0.1724
0.2213
0.3402
0,6563
0.7288
No Physical Loss
4.0092
0.3569
0.0674
0.1541
0.2081
0.3274
0.6547
0.7299
PINP (this work)
4.1684
0.3800
0.0989
0.1952
0.2448
0.3573
0.6556
0.7279
E.4 PHYSICAL LOSS ANALYSIS ON THE SEVIR DATASET.
The nowcasting scenario may not strictly adhere to the NS equations. However, we believe that incorporating the NS equations helps to capture the motion characteristics of nowcasting.
conducted the following experiments: (1) Lowered the physical loss weight (weight = 0.1) while keeping other settings unchanged. (2)Removed the physical loss entirely.
The results are showned on Table S9, which demonstrate that while nowcasting processes not strictly adhere to the NS equations, these equations effectively capture certain underlying motion characteristics. When the physical loss weight is reduced, loosening the constraints, the MSE de- creases. However, this relaxation comes at the cost of reduced ability model essential motion features, as reflected in the decline of the CSI.
F ADDITIONAL RESULTS
In this section, we provide additional details on the experimental results presented the main text. This includes supplementary results for multi-physical quantity inference (Figure S9, Figure S10), a comparison between our predicted velocity field and the velocity field predicted by NowcastNet Figure S11, detailed comparison between our method and baselines the 2D fluid data (Figure S12, Figure S13), 2D smoke data (Figure S S1), 3D smoke data (Figure S15, Figure S S16), and SE VEVIR data (Figure $17, Figure S18).
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Replacing s(l") with c(t)
Real
f2
Byy
20
30
40
Figure S8: example showing the results after replacing c(t') with c(t).
10
20
30
t= 40
Real
4,
Figure S9: An example of 2D fluid prediction and latent physical quantity inference.
Real
Real
20
R
1-48
2k
C
111
Figure S10: An example of 2D smoke prediction and latent physical quantity inference.
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Real
PINP
NowcastNet
PySTEPS
Figure S11: The Comparison Between Our Predicted Velocity Field and the Velocity Field Predicted by NowcastNet and PySTEPS. The black regions in PySTEPS represent NaN values.
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I
I
Prodiction Error
Prediction
Prediction
Brror
-aso -aus
Prediction Ereor
-235
Figure S12: An example comparing our model with the baseline on fluid data.
Real
U-NO
U-FNO
LSM
FNO
PINP
Prodiction Ereor
Prediction
Prediction Error
kc
Prediction Frror
Figure S13: An example comparing our model with the baseline on 2D fluid data.
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Real
PINP
HelmFluid
U-FNO
U-NO
Prediction Error
0
2
2
Prediction Error
-2
Prediction
Error
Prediction Error
0
-2
100
Prediction
Figure S14: An example comparing our model with the baseline 2D smoke data.
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t=2
t=4
t=
t=8
[= 10
2.00
1.75
Real
1.50
1.25
1.00
FNO
0.00
UNO
UFNO
LSM
HelmFluid
PINP
Figure S15: An example comparing our model with the baseline on 3D smoke data.
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t=2
6=6
Real
1.36
1.25
1.00
0.75
FNO
0.5C
UNO
UFNO
LSM
HelmFluid
PINP
Figure S16: An example comparing our model with the baseline on 3D smoke data.
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mön
min
30
D-16
16:31
59-74
100-133
133-160
181-219
ComvL.STM
219-255
PredRNN
SimVP
Earthformer
NowCastNer
PINP
Figure S17: An example comparing our model with other baselines in nowcasting.
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minn
min
30 min
D-16
16-31
Real
59-74
100-133
133-160
181-219
219-255
PredRNN
SimVP
Earthformer
NawCastNet
PINP
Figure S18: An example comparing our model with other baselines in nowcasting.
31


Reviews:
[{"review_id": "7fWcjdR1yl", "reviewer": "Reviewer_zzB8", "summary": "This paper proposed the PINP method for fluid prediction. The method predicts future fluid fields by learning velocity and pressure simultaneously from partial observations. The authors employ a physical inference neural network to predict several physical quantities of the flow field at a particular moment. For the next timestep, they utilizes discrete PDEs predictor and correction network to generate the flow field. The training of the model is refined through the application of MSE loss, equation loss, and a temporal constraint loss. The proposed method shows advantages compared to several baselines on both synthetic and real-world data.", "questions": "- For the sole real dataset, SEVIR, raises several concerns:\n  - It is unclear why the MSE metric for the proposed method underperforms compared to most other baselines. A detailed analysis and discussion of this discrepancy would be beneficial.\n  - Given the potential phase transition of water in the SEVIR dataset, it is questionable whether the fluid satisfies the incompressible property. The authors are encouraged to discuss the applicability of the equation loss function to real datasets with such characteristics.\n  - I highly recommend the authors compare with the traditional numerical method pySTEPS[1], which predicts future fluid fields by estimating potential velocity fields and extrapolating optical flow. This method has the ability to accurately estimate extreme values.\n- The paper does not specify the fluid's Reynolds number, which is crucial for understanding the flow characteristics. The fluid in the Fluid 2D dataset appears to represent simple laminar flows. The authors are recommended to provide experiments with more turbulent datasets to enhance the paper's practical value.\n- The paper lacks information on how the baselines were trained, and some baselines exhibit abnormal flickering. The authors should recheck the training process for all baselines or explain these anomalies.\n- The visualization of velocity in the paper is not as intuitive as it could be. Utilizing tools such as *pyplot.quiver* in *matplotlib* to depict the velocity field based on flow field observations as supplementary material is suggested.\n- The grammar and expression throughout the paper should be improved. A thorough review and enhancement are advised.\n\n\n[1] https://pysteps.github.io/", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "This paper combines equation loss with operator learning through the Navier-Stokes equation, integrating physics-driven and data-driven approaches by learning unobserved physical quantities.\n- An innovative point of the proposed method is incorporating the equation loss commonly used in PINN, including incompressible Navier-Stokes equations, into predicting potential physical quantities of the future flow field.\n- The figures provided in the paper can accurately reflect the characteristics of the model.\n- The ablation of the paper accurately explains the role played by each module.", "weaknesses": "This paper has some weaknesses, including:\n- Some datasets are too simple or may theoretically not match the methods to some extent.\n  - The fluid motion in the Flow 2D dataset is relatively slow, and the dynamics are not as complex as those encountered in more advanced fluid dynamics scenarios.\n  - The fluids in real datasets may not strictly adhere to the incompressible Navier-Stokes equations. Consequently, the physical constraints proposed in this paper might encounter limitations when applied to more diverse or complex fluid systems.\n- The improvements observed in specific datasets, such as Smoke3D, are modest. For example, the MAE and MSE metrics of Smoke 3D only demonstrate a marginal enhancement.\n- The paper's grammar and expression could be improved. In some instances, the clarity of the writing detracts from the overall quality of the paper, potentially hindering the reader's understanding of the research.\n\nSpecific issues and possible improvements will be discussed in the next section.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "voOIZVIzwY", "reviewer": "Reviewer_m8WU", "summary": "This work presents a new physics-informed learning approach that enables the prediction of coupled physical states, under a partially observed data environment. It applies the discretization of physical equations, integrating into the model architecture and loss function. The superior performance is shown in four benchmarks including a real-world data.", "questions": "Please check the weaknesses.", "limitations": "", "rating": 6, "confidence": 5, "soundness": 3, "presentation": 2, "contribution": 3, "strengths": "1. The proposed method enhances the simulation capacity of PDEs, especially on the long-term prediction.\n2. The proposed method is tested on multiple benchmarks across different scenarios, especially on the real-world measured dataset.\n3. The authors vividly demonstrated the simulation process through videos.", "weaknesses": "1. The reviewer believes that there are significant issues with the introduction of the method. The method is complex and lacks an overview of the proposed method. The reviewer understands that the proposed method first outputs $p(t')$, $u(t')$, and $c(t')$ through a physical inference network, where these three outputs are constrained by physical and temporal conditions. Then $\\hat{c}'(t+1)$ is computed through discretized PDEs, after which $\\hat{c}'(t+1)$ and $c(t)$ are fed into another network for prediction, while simultaneously incorporating a data loss with the label. If this understanding is correct, the reviewer questions the novelty of this work, as it merely sandwiches numerical FDM calculations between neural networks. Moreover, the motivation for this approach remains unclear.\n2. In line 039, the statement is inaccurate and needs to be referenced to the literature. The reviewer points out that velocity fields can be observed through techniques such as PIV and PTV.\n3. In line 069, what does the past observable data mean. Authors should introduce more about the settings.\n4. In line 102, \"often difficult to obtain in practical applications\". The reviewer considers the statement is inappropriate, as initial conditions are typically obtainable when solving PDEs.\n5. In Table 1, the reviewer appears to have misinterpreted the meaning of the three categories in this table. If 'velocity' refers to velocity fields, then this table is not appropriate, as FNO (Fourier Neural Operator) is equally capable of predicting both velocity and pressure fields.\n6. In Eqn. 3, why does this equation still integrate from t to t+1? A more detailed derivation process is needed to help readers understand. This is crucial for comprehending the motivation behind the problem. What is the meaning of $\\Delta t$?\n7. In Sec. 3.4, the introduction is oversimplified, merely stating which networks are used. This raises two concerns for the reviewer: first, why was U-Net chosen over more advanced transformer architectures, and second, too many network structural details are omitted, forcing readers to consult the appendix for understanding.\n8. In Sec. 3.5, author should carfully introduce the training process as there are many networks and parameters. Are they trained in an end-to-end manner? This raises a question about how the physials inference network can simultaneously learn Pe and output flow fields. These two components might interfere with each other, potentially making the network untrainable. Has the use of stop-gradient operations like VQVAE been considered?\n9. What is the PDE for the real-world data? Is it explicitly known? Real data often comes with noise - has this method considered noise effects, or are there any approaches proposed to address the influence of noise?\n10.  In Sec. 5, especially in Fig. 9(a), the authors need to specify the number of experimental trials conducted and report the confidence levels, as it appears that the two constraints overlap for an extended period of time.\n11. What is the detail setting of the fluid 2D data, including $\\nu$, $dx$, $dt$, and boundary conditions?\n12. The reviewer does not find the link of code and dataset from the paper. Code and data are important criteria for verifying the rationality of results. Will the author make them opensource？", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "fdTqL0Vw5g", "reviewer": "Reviewer_tKs8", "summary": "This paper's core idea is to combine a data-centric deep learning approach with physics by incorporating the discretised Navier-Stokes equations into the neural network architecture and constraining the loss function. By explicitly incorporating the governing equations and the associated physical quantities, the authors try to model the system and help with consistency, interpretability, and extrapolation capabilities. The extensive proposed experiments show good performances and generalisation to unseen domains.", "questions": "- The gradient discretisation used here is a second-order central difference approximation. Could you elaborate on whether there were specific architectural reasons for choosing this method over other discretisation schemes? Additionally, it would be helpful to understand if you considered alternative discretisation approaches\nYou compare two different sets of baseline: one for now-casting and one for Navier-Stoke simulation. Why is your model capable of doing both, as other neural operator-based models are not? That seems like a significant advantage that has yet to be developed.", "limitations": "", "rating": 8, "confidence": 3, "soundness": 4, "presentation": 3, "contribution": 3, "strengths": "- The paper is very well written and explains complex problems clearly.\n- While the idea of incorporating PDE-based constraints in the network architecture and loss function is not new, the author presents a set of methods, tricks, and ideas that make the technique work better than previous literature.", "weaknesses": "- Interpretability depends on the other physical quantities' models. While there are theoretical reasons to believe the quantities are interpretable, there is little experimental evidence.\n- The pertinence of benchmarking nowcasting and the advantage of this method over other neural operator-based methods for this task is unclear.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "Xpsa4bLaJj", "reviewer": "Reviewer_3QEU", "summary": "This paper proposes a new physics-informed fluid predictor, named PINP. PINP firstly estimates the underlying pressure and velocity filed from observed fluid, which is constrained by a discretized physics loss. Then it employs an interpolation formalization of integral for future prediction, where an additional correction network is presented to reduce the error of discretized PDE predictor. Experimentally, PINP performs well in 2D and 3D flows and weather prediction tasks.", "questions": "1.\tAbout implementation of baselines.\n\nIn NowCastNet, ensuring eidetic prediction results is one significant contribution of this paper. However, as shown in Figure 8, its prediction is quite blurry. I am wondering how the authors experimented with this baseline.\n\nBesides, in the supplementary materials, the prediction results of LSM and FNO appear strange periodic shakes. Actually, I think a well-trained deep model will not make such weird predictions. Did the authors carefully tune these two baselines?\n\n2.\tAbout spatial generalization.\n\nWhy PINP can achieve spatial generalization? Can the authors provide some intuitive explanations?", "limitations": "", "rating": 6, "confidence": 4, "soundness": 2, "presentation": 3, "contribution": 2, "strengths": "This paper is overall well-written.\n\nThe idea of incorporating physics loss into fluid prediction is reasonable.\n\nThe authors have provided comprehensive experiments to verify the effectiveness of the proposed method.", "weaknesses": "1.\tThe title is kind of overclaimed. \n\n\nSince this paper is tailored to fluid prediction, I think “physics-informed fluid predictor” is more suitable. Otherwise, it is a little bit overclaimed, where there are extensive prediction tasks that PINP cannot solve, such as rigid body movement (controlled by classical physics) or magnetic field (governed by electromagnetism).\n\n2.\tA series of technical designs are underexplored or not well supported.\n\n(1)\tPINP adopts the discretized PDE loss for physics constraint, which may bring serious approximation error. The current design is based on the assumption that the differential operator can be approximated by spatial or temporal difference, which cannot be satisfied, especially in low-resolution data. Note that I am not saying that being physics-informed is a bad idea. The canonical physics-informed neural works employ the auto differential in neural works for approximation, which is much more precise than the discretization in PINP.\n\n(2)\tI cannot figure out that why additionally predicting the pressure field can boost the performance. As shown in Figure 2 (b), the predicted pressure field is only used in physical constraint loss, which cannot affect the future prediction process. This means that predicting the pressure field is just to fit the physical loss, which brings a new meaningless task. According to my experience, I think this design can only bring extra load to the model instead of benefiting the prediction. Besides, as shown in Figure 9(a), removing physical constraints will not bring a serious decrease. Further, How about keeping the second equation in Eq.(12) but removing the pressure-related one? I believe that the benefit of physics loss is mainly brought by the incompressible term loss. \n\n(3)\tThe design of the correction network is also weird. As formalized in Eq.(10), the inputs and outputs of the correction network are both expected to be close to the ground truth. Under this constraint, why correction network is necessary? (Minor: Eq.(10) may have a typo, where the comma should be “-”).\n\n(4)\tAbout the temporal loss. I am curious about how likely is this loss function to work. Some statistical results on how many times this loss is non-zero are expected.\n\nGoing further from (2), I doubt that the prediction of pressure field is useless in the current design, which is listed as one of the main contributions w.r.t. other papers. I think compared with Helmfluid, the advantage of PNIP lies in the physical loss, which can provide a more direct and explicit constraint to the velocity field.\n\nIn summary, I think there are many unsupported designs in the proposed method, which may affect the claim of the main contribution of this paper.\n\n3.\tAbout the efficiency. \n\nI am curious about the training overload. Since the calculation of loss in Eq.(16) may also cause extra computation costs than other baselines.", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Decision: Accept (Poster)